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Talk:System of equations

General structure of article
I came to this article because I was willing to write an article (which is lacking) on polynomial systems of equations. For this it was natural to start by reference to articles defining the basic facts like "What is an equation?", "What means solving an equation or a system of equations?". I have been unable to find any article dealing to these questions, the present one being the one which is the closest from what I need.

The article is implicitly restricted to a particular set of equations, namely those whose unknown values have to be found in a field. This excludes many important classes of equations and systems of equations: boolean equations, functional equations, differential and integral equations, Diophantine equations,...

My suggestion, which I am unable to realize myself because of lack of time and of experience of Wikipedia: D.Lazard (talk) 16:50, 19 May 2010 (UTC)
 * To write an article entitled "Equations and systems of equations" which answers to above questions and present the main classes of equations and systems, with links to articles devoted to them.
 * To write articles on
 * Linear systems. The present article could be a starting point, as the methods which are presented here (substitution and elimination) work generally only in this case. By the way, they are both essentially equivalent to Gaussian elimination.
 * Multivariate polynomial systems: I am willing to write it, at least the part devoted to algebraic methods.
 * Transcendental equations and systems
 * and many other

Material moved out of another page
I have removed the following from the page Substitution method, which deals with an optical technique. I was going to create a new page for this, but perhaps it belongs here or in some other existing article. I'll leave it up to the math folks to figure out where this best belongs--Srleffler 03:51, 11 March 2006 (UTC):

The substitution method is an algebraic method for solving a system of equations (finding the point where two graphed lines intersect). The substitution method, unlike the elimination method, will solve for any type of system, whereas the elimination method will only solve for linear systems. In the substitution method, you do not have to have the equations in the same form. The substitution method infers that since the same variables are used, they equal the same thing. For example, for functions, two y's in an equation, although they may differ in the terms that they are said to be dependent upon, must equal the same thing. Take the equations y = x and y = 2x - 10. Since the same variables are used, they can be easily substituted. In this set of circumstances, you should take one equation (y = x for here), and substitute it in for the y in the other equation. You receive: x = 2x - 10. Then, simply solve it for x. Since there is a difference of 10 between the two, you should receive x = 10. Then, take the other equation (y = 2x - 10), and substitute 10 in for the x's. You should receive: y = 2(10) - 10. This simplifies to 20 -10 = y. Therefore, y = 10. The solution for the system is the ordered pair (10, 10). Always remember to substitute into DIFFERENT equations when you have solved for the first variable. You will not receive a correct answer if you substitute into the same equation.

==References==

Expert needed
If you're an expert you'll see for yourself this article has "much room for improvement". --Lambiam Talk 15:26, 26 September 2006 (UTC)
 * 1) It is fine that it starts with an example, but then it should also give the solution for the example in the introductory paragraph, where perhaps something more elementary than the present example is better -- such as (x+2y = 7 & xy = 6) with solutions (x = 3 & y = 2) and (x = 4 & y = 1.5).
 * 2) The intro should not immediately talk about a geometric interpretation. If this is done (later), it should be explained what the relation is between these equations and geometry. This is mostly useful only when there are two or perhaps three real-valued unknowns, but generally not for four unknowns or two complex unknowns.
 * 3) There should be some clarification of the domain over which the unknowns can range. Is this a system of Diophantine equations? Are we constrained to real numbers or can we have complex solutions? Some other domain? We could also have a system of differential equations, in which the unknowns are functions.
 * 4) The discussion between the # of equations and the # of variables is incomplete and partly wrong. What does it mean that "every variable will have an explicit solution set"? Aren't we solving simultaneously? What is the meaning of "explicit" here? Why isn't this the case when y>x (except for the finiteness)? Examples are needed here. There should be some discussion of the possibility that there is no solution at all. The claim as stated is "somewhat truish" for the real & complex domains, but not at all for integers.
 * 5) It should be made clear that there exists no general method for solving equations -- let alone simultaneous equations.
 * 6) It is not true that systems of simultaneous linear equations "can always be solved" if that means: we can always find solutions. This false interpretation may be the "obvious" interpretation of this sentence for most innocent readers.
 * 7) On the other hand, elimination is sometimes possible for non-linear equations. If x = r cos φ and y = r sin φ, then x2 + y2 = r2, eliminating φ. Likewise, x/y = tan φ, eliminating r.
 * 8) Numerical solving methods should be a separate section. They are not a special case of el;imination.
 * 9) The penultimate sentence does not belong here.
 * 10) The last sentence also does not fit the topic "elimination". This has some relation to Ansatz methods.
 * 11) There are many more methods which sometimes may be succesful. A few examples would not hurt.

 * Kay, I'm taking a look at it. I'll try and implement the requests you've listed here. --Brad Beattie (talk) 13:51, 19 November 2006 (UTC)

The cyclic method is mental math. Set up the difference of cross-products for the numerator of the value of each variable and another difference of cross-products for the denominator. Larry R. Holmgren 18:56, 23 March 2007 (UTC)

Although there is an example of a linear equation and a second degree equation (circle) other examples are systems of linear equations. Could we add a section on solving systems of quadratic equations. There are more steps but methods are similar, multiple substitutions and elimination of a variable. Larry R. Holmgren 04:03, 24 March 2007 (UTC)

Additionally, constrained systems of equations can be handled with a Lagrange multiple. I could add these three sections using equations of polynomials. The article on Lagrange multipliers does not have such a section.

Yes, could someone else contribute systems of differential equations? An introduction would be good. The article on differential equations does not solve any, nor does it cover systems of equations. Larry R. Holmgren 04:10, 24 March 2007 (UTC)

 * "It is fine that it starts with an example, but then it should also give the solution for the example in the introductory paragraph, where perhaps something more elementary than the present example is better -- such as (x+2y = 7 & xy = 6) with solutions (x = 3 & y = 2) and (x = 4 & y = 1.5)."
 * I did that.
 * "The intro should not immediately talk about a geometric interpretation. ..."
 * I did that.
 * Does anyone dispute my work? Just wondering ~user:orngjce223 how am I typing? 23:35, 12 October 2007 (UTC)

Merging Suggestion
I am proposing a merging of this article with part of Elementary algebra's info on Systems of Linear Equations and System of linear equations. Comments are appreciated and the explanation and discussion are being held here: Wikipedia talk:WikiProject Mathematics. (Quadrivium 23:17, 17 November 2006 (UTC))

Yes, the topics are the same and should be merged. Systems of Linear Equations is set up for a solution by determinants (matricies) which would have computer applications in statistics and econometrics and would be added to part 1.3 of this article, whereas Simultaneous equations is just a first year algebra topic, second semester. Larry R. Holmgren 19:06, 23 March 2007 (UTC)

No. As a schoolboy I've just come to this page to find out about these equations and it's been a great help and very well explained. I found it easily from Google. I don't want to get bogged down in any other topic. Leave it alone. 86.31.78.115 19:41, 11 April 2007 (UTC)

 * Is the article intended to be only about systems of simultaneous linear equations? An editor has just put a new tag "Merge with System of linear equations" on the article, and has next replaced all examples of non-linear equations in the article by linear equations. I find this a curious way of operation; it is like proposing to merge Religion with Buddhism and then proceeding, without awaiting the discussion, to erase all references to other belief systems than Buddhism from the Religion article. --Lambiam 22:51, 19 March 2008 (UTC)
 * You're quite right. The distinction between the subjects is important, and there is so much to say about the linear case that a merged article would (appear to) have the general case as a footnote.  I've been bold; I struck the merge tag (which was added 16 months after this talk section was started) and restored the worked-out non-linear example.  (The linear example wasn't correctly solved anyway.)  --Tardis (talk) 16:56, 17 April 2008 (UTC)

 * I propose a different action. I think this article, as well as System of linear equations, should be moved to System of equations, as this is the more common term used in academia (or for that matter, the term used at my school). Cheers, The Doctahedron, 23:09, 7 January 2012 (UTC)

Support with condition: System of equations already redirects here. I think that this article has to be rewritten into the following lines: After the lead, a description of the elementary methods. Then sections on the specialized and advanced methods, with template {main}: System of linear equations, system of polynomial equations, system of ordinary differential equations, ... (note that system of ordinary differential equation, which is grammatically incorrect, redirects to ordinary differential equation, which does not talk specifically on systems) The sections least-squares and matrices should either be suppressed or merged in the convenient specialized page. D.Lazard (talk) 11:57, 8 January 2012 (UTC)

 * Procedural oppose. The above template was added to a long-dead thread.  It's not clear at all what is being voted on.  I propose that this be marked as speedily declined, and a proper proposal and thread can be started.   Sławomir Biały  (talk) 12:44, 8 January 2012 (UTC)
 * The template was removed. If anyone really wants to rename this article, then feel free to start a discussion under a new heading but not above the current discussion. Vegaswikian (talk) 21:52, 8 January 2012 (UTC)

Cyclic Rule
I've reverted the addition of a section on this method from so-called "Vedic mathematics". Among the problems I see are the following (not necessarily in order of importance): Perhaps an improved version belongs in the article on Swami Bharati Krishna Tirtha's Vedic mathematics, or tucked away in a section on "Other solution methods" in System of linear equations. --Lambiam Talk 08:14, 27 March 2007 (UTC)
 * There is much wrong or confusing about the present article, which should be fixed before more confusing material is added.
 * This method is rather obscure and of unclear importance, and does not belong in an entry-level class article.
 * The description of the method is difficult to follow and unclear for me as a trained mathematician, and presumably fairly incomprehensible for the reader for whom the article is written.
 * The description does not set the context. To which kinds of systems of equations does it apply? Not to {5x − 4y + z + 8 = 0, 7x2 − 3y2 + 3z2 + 9 = 0, − 5y + z2 + 10 = 0}.
 * There is no mathematics markup of any kind.

Inequalities
Simultaneous equations may well involve inequalities. This relates to the area of Linear Programming and Operations Research.--Billymac00 00:57, 15 September 2007 (UTC)

Matrices
This section of the article is unintelligible. It needs to be written in much clearer language with simple worked examples. This explanation is useless to anyone who does not already know the method. —Preceding unsigned comment added by 81.157.45.30 (talk) 15:44, 6 August 2009 (UTC)

Number of solutions
(Moving this discussion into a new section) D.Lazard (talk) 12:40, 16 May 2011 (UTC)

If the number of equations is the same as the number of variables, then probably (but not necessarily) the system is exactly solvable in the sense that the set of its solutions is infinite.

Should it really be infinite here, or finite? This would correspond more to my intuition of exactly solvable. —Preceding unsigned comment added by 193.190.253.144 (talk) 19:30, 6 February 2010 (UTC)

Talk:Simultaneous equations/rewrite While the initial article about "Simultaneous equation" (since moved to "Simultaneous equations" and rewritten) was indeed impenetrable, statisticians do consider "Simultaneous equation models". http://www.google.com/search?q=site%3A.edu+simultaneous+equation+endogenous+Hausman gives 750 hits. AxelBoldt 23:00 20 Jun 2003 (UTC)

If there are fewer equations than variables, there may be infinitely many solutions; if there are more equations than variables, there may be no solution.

This is misleading: x=x, x+1=x+1 has infinitely many solutions, x+y=x+y+1 has none; reverted. - Patrick 02:09 21 Jun 2003 (UTC)

1) Yeah. The key words are "may have", I think. and 2) x+y=x+y+1 is not an equation.

Re:1) Disagree: A key word is "independent" - the many equations must be independent, otherwise one equation is just another equivalent form of a previous equation. For example, x=x and x+1=x+1 are not independent equations because if you add a constant to both sides of the second equation (which you are allowed to do in algebra and it doesn't change the equality) then you wind up with the first equation. E.g., add the constant -1 (minus one) to both sides, and the result is x=x. So x=x and x+1=x+1 are not two separate equations, they are really the same one. 71.125.150.190 (talk) 17:54, 14 May 2011 (UTC)

 * Independent is clearly an important notion. Unfortunately, in the non linear case, testing independence is usually as difficult as solving. This matter is discussed in more details in article system of polynomial equations. Here under determined and over determined systems are considered, if the number of equations is respectively lower or higher than the number of unknowns. Over determined systems have generally no solution (inconsistent system), but may have some solutions if some hidden dependence relations occur. Under determined have generally an infinite number of complex solutions but may be inconsistent, like x+y = x+y+1 = 0. In the case of linear or polynomial systems, under determined systems have either an infinite number of complex solutions or no solution. But this is not true if only real solutions are considered or if the equations are not polynomial.

 * I agree that this article should be made more accurate, but it is difficult to do this, remaining at the low mathematical level needed for this article. Someone is willing to do this? D.Lazard (talk) 13:07, 16 May 2011 (UTC)

WP:SELFPUBLISH?
I cant see any citation,reference from reliable source except a webpage from Simon Bruce. I think Simultaneous equations or system of equations appear in Wikipedia thanks to WP:COMMONNAME, but in turn, seems to expand to article like Autonomous system, Nonlinear system, System of linear equations for example, see also Category:Systems theory and Portal:Systems science. The problem is the semi-mathematical and semi-pseudoscience properties of these article, we need discussion on whether they should be removed, rephrased or moved away from Mathematics Portal. The word system seems to be informal and ambiguous in Mathematics(WP:NATURALDIS) but it is quite common on other field(WP:COMMONNAME). Imo, this shouldn't be considered and can't be qualified as a project in Mathematics Portal due to the lack of verifiability or even notability within Mathematics community. In mathematics we often talk about family of sets and collection of objects, if there is some properties there we have tons load of algebraic structure like category, group. On the other hand, from the example articles given above, we can see they all suffer from lack of verifiability, and they lack section on precise definition on what a system is. They do mention the word system but evade the definition part, all wiki to this article which is in fact suffering from WP:SELFPUBLISH. --14.198.221.19 (talk) 09:14, 30 March 2013 (UTC)

 * This article has multiple issues already described in this talk page, but not those of the preceding post: This article lacks of citations, but does not contain self published nor WP:original research; references for is content may be found in many places, inside (see, for example system of linear equations, elementary algebra and other targets of the links of this article) and outside WP. I do not know what is "semi-mathematical" and "semi-pseudoscience", but these term can not qualify this article, which, although elementary, is pure mathematics and thus pure science. "System" is ambiguous in mathematics (and not only there), but "system of equations" is not: this is a terminology that is well established since 19th century, much before the rise of "system theory" and "system science". Beside of the lack of citation, the main issue of this article is that (except for the lead), its content duplicates system of linear equations and, partly, elementary algebra and that it does not consider other types of systems of equations. IMO, the best way to solve these issues is to transform these article in a WP:DAB page. I'll be WP:bold and edit it in this way. D.Lazard (talk) 07:25, 3 April 2013 (UTC)

title: Talk:System of equations


text:
Simultaneous equations model

Simultaneous equations models are a type of statistical model in which the dependent variables are functions of other dependent variables, rather than just independent variables. This means some of the explanatory variables are jointly determined with the dependent variable, which in economics usually is the consequence of some underlying equilibrium mechanism. Take the typical supply and demand model: whilst typically one would determine the quantity supplied and demanded to be a function of the price set by the market, it is also possible for the reverse to be true, where producers observe the quantity that consumers demand and then set the price.

Simultaneity poses challenges for the estimation of the statistical parameters of interest, because the Gauss–Markov assumption of strict exogeneity of the regressors is violated. And while it would be natural to estimate all simultaneous equations at once, this often leads to a computationally costly non-linear optimization problem even for the simplest system of linear equations. This situation prompted the development, spearheaded by the Cowles Commission in the 1940s and 1950s, of various techniques that estimate each equation in the model seriatim, most notably limited information maximum likelihood and two-stage least squares.

Structural and reduced form
Suppose there are m regression equations of the form

y_{it} = y_{-i,t}'\gamma_i + x_{it}'\;\!\beta_i + u_{it}, \quad i=1,\ldots,m, $$

where i is the equation number, and is the observation index. In these equations xit is the ki×1 vector of exogenous variables, yit is the dependent variable, y−i,t is the ni×1 vector of all other endogenous variables which enter the ith equation on the right-hand side, and uit are the error terms. The “−i” notation indicates that the vector y−i,t may contain any of the y’s except for yit (since it is already present on the left-hand side). The regression coefficients βi and γi are of dimensions ki×1 and ni×1 correspondingly. Vertically stacking the T observations corresponding to the ith equation, we can write each equation in vector form as

y_i = Y_{-i}\gamma_i + X_i\beta_i + u_i, \quad i=1,\ldots,m, $$ where yi and ui are T×1 vectors, Xi is a T×ki matrix of exogenous regressors, and Y−i is a T×ni matrix of endogenous regressors on the right-hand side of the ith equation. Finally, we can move all endogenous variables to the left-hand side and write the m equations jointly in vector form as

Y\Gamma = X\Beta + U.\, $$ This representation is known as the structural form. In this equation is the T×m matrix of dependent variables. Each of the matrices Y−i is in fact an ni-columned submatrix of this Y. The m×m matrix Γ, which describes the relation between the dependent variables, has a complicated structure. It has ones on the diagonal, and all other elements of each column i are either the components of the vector −γi or zeros, depending on which columns of Y were included in the matrix Y−i. The T×k matrix X contains all exogenous regressors from all equations, but without repetitions (that is, matrix X should be of full rank). Thus, each Xi is a ki-columned submatrix of X. Matrix Β has size k×m, and each of its columns consists of the components of vectors βi and zeros, depending on which of the regressors from X were included or excluded from Xi. Finally, is a T×m matrix of the error terms.

Postmultiplying the structural equation by Γ−1, the system can be written in the reduced form as

Y = X\Beta\Gamma^{-1} + U\Gamma^{-1} = X\Pi + V.\, $$ This is already a simple general linear model, and it can be estimated for example by ordinary least squares. Unfortunately, the task of decomposing the estimated matrix into the individual factors Β and Γ−1 is quite complicated, and therefore the reduced form is more suitable for prediction but not inference.

Assumptions
Firstly, the rank of the matrix X of exogenous regressors must be equal to k, both in finite samples and in the limit as T → ∞ (this later requirement means that in the limit the expression should converge to a nondegenerate k×k matrix). Matrix Γ is also assumed to be non-degenerate.

Secondly, error terms are assumed to be serially independent and identically distributed. That is, if the tth row of matrix U is denoted by u(t), then the sequence of vectors {u(t)} should be iid, with zero mean and some covariance matrix Σ (which is unknown). In particular, this implies that E[U] = 0, and E[U′U] = T Σ.

Lastly, assumptions are required for identification.

Identification
The identification conditions require that the system of linear equations be solvable for the unknown parameters.

More specifically, the order condition, a necessary condition for identification, is that for each equation $k_{i} + n_{i} ≤ k$, which can be phrased as “the number of excluded exogenous variables is greater or equal to the number of included endogenous variables”.

The rank condition, a stronger condition which is necessary and sufficient, is that the rank of $Π_{i0}$ equals $n_{i}$, where $Π_{i0}$ is a $(k − k_{i})×n_{i}$ matrix which is obtained from $Π$ by crossing out those columns which correspond to the excluded endogenous variables, and those rows which correspond to the included exogenous variables.

Using cross-equation restrictions to achieve identification
In simultaneous equations models, the most common method to achieve identification is by imposing within-equation parameter restrictions. Yet, identification is also possible using cross equation restrictions.

To illustrate how cross equation restrictions can be used for identification, consider the following example from Wooldridge

 * $$\begin{align}

y_1 &= \gamma_{12} y_2 + \delta_{11} z_1 + \delta_{12} z_2 + \delta_{13} z_3 + u_1 \\ y_2 &= \gamma_{21} y_1 + \delta_{21} z_1 + \delta_{22} z_2 + u_2 \end{align}$$

where z's are uncorrelated with u's and y's are endogenous variables. Without further restrictions, the first equation is not identified because there is no excluded exogenous variable. The second equation is just identified if $δ_{13}≠0$, which is assumed to be true for the rest of discussion.

Now we impose the cross equation restriction of $δ_{12}=δ_{22}$. Since the second equation is identified, we can treat $δ_{12}$ as known for the purpose of identification. Then, the first equation becomes:

 * $$y_1 - \delta_{12} z_2 = \gamma_{12} y_2 + \delta_{11} z_1 + \delta_{13} z_3 + u_1$$

Then, we can use $(z_{1}, z_{2}, z_{3})$ as instruments to estimate the coefficients in the above equation since there are one endogenous variable ($y_{2}$) and one excluded exogenous variable ($z_{2}$) on the right hand side. Therefore, cross equation restrictions in place of within-equation restrictions can achieve identification.

Two-stage least squares (2SLS)
The simplest and the most common estimation method for the simultaneous equations model is the so-called two-stage least squares method, developed independently by and. It is an equation-by-equation technique, where the endogenous regressors on the right-hand side of each equation are being instrumented with the regressors X from all other equations. The method is called “two-stage” because it conducts estimation in two steps:
 * Step 1: Regress Y−i on X and obtain the predicted values ;
 * Step 2: Estimate γi, βi by the ordinary least squares regression of yi on and Xi.

If the ith equation in the model is written as

y_i = \begin{pmatrix}Y_{-i} & X_i\end{pmatrix}\begin{pmatrix}\gamma_i\\\beta_i\end{pmatrix} + u_i \equiv Z_i \delta_i + u_i, $$ where Zi is a T×(ni + ki) matrix of both endogenous and exogenous regressors in the ith equation, and δi is an (ni + ki)-dimensional vector of regression coefficients, then the 2SLS estimator of δi will be given by

\hat\delta_i = \big(\hat{Z}'_i\hat{Z}_i\big)^{-1}\hat{Z}'_i y_i = \big( Z'_iPZ_i \big)^{-1} Z'_iPy_i, $$ where is the projection matrix onto the linear space spanned by the exogenous regressors X.

Indirect least squares
Indirect least squares is an approach in econometrics where the coefficients in a simultaneous equations model are estimated from the reduced form model using ordinary least squares. For this, the structural system of equations is transformed into the reduced form first. Once the coefficients are estimated the model is put back into the structural form.

Limited information maximum likelihood (LIML)
The “limited information” maximum likelihood method was suggested by M. A. Girshick in 1947, and formalized by T. W. Anderson and H. Rubin in 1949. It is used when one is interested in estimating a single structural equation at a time (hence its name of limited information), say for observation i:

 * $$  y_i = Y_{-i}\gamma_i  +X_i\beta_i+ u_i \equiv Z_i \delta_i + u_i  $$

The structural equations for the remaining endogenous variables Y−i are not specified, and they are given in their reduced form:
 * $$  Y_{-i} = X \Pi + U_{-i} $$

Notation in this context is different than for the simple IV case. One has:
 * $$Y_{-i}$$: The endogenous variable(s).
 * $$X_{-i}$$: The exogenous variable(s)
 * $$X$$: The instrument(s) (often denoted $$Z$$)

The explicit formula for the LIML is:

\hat\delta_i = \Big(Z'_i(I-\lambda M)Z_i\Big)^{\!-1}Z'_i(I-\lambda M)y_i, $$ where, and λ is the smallest characteristic root of the matrix:

\Big(\begin{bmatrix}y_i\\Y_{-i}\end{bmatrix} M_i \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} \Big) \Big(\begin{bmatrix}y_i\\Y_{-i}\end{bmatrix} M \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} \Big)^{\!-1} $$

where, in a similar way,.

In other words, λ is the smallest solution of the generalized eigenvalue problem, see :

\Big|\begin{bmatrix}y_i&Y_{-i}\end{bmatrix}' M_i \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} -\lambda \begin{bmatrix}y_i&Y_{-i}\end{bmatrix}' M \begin{bmatrix}y_i&Y_{-i}\end{bmatrix} \Big|=0 $$

K class estimators
The LIML is a special case of the K-class estimators:

\hat\delta = \Big(Z'(I-\kappa M)Z\Big)^{\!-1}Z'(I-\kappa M)y, $$

with: Several estimators belong to this class:
 * $$ \delta = \begin{bmatrix} \beta_i & \gamma_i\end{bmatrix} $$
 * $$ Z = \begin{bmatrix} X_i & Y_{-i}\end{bmatrix} $$
 * κ=0: OLS
 * κ=1: 2SLS. Note indeed that in this case, $$ I-\kappa M = I-M= P $$ the usual projection matrix of the 2SLS
 * κ=λ: LIML
 * κ=λ - α / (n-K): estimator. Here K represents the number of instruments, n the sample size, and α a positive constant to specify. A value of α=1 will yield an estimator that is approximately unbiased.

Three-stage least squares (3SLS)
The three-stage least squares estimator was introduced by. It can be seen as a special case of multi-equation GMM where the set of instrumental variables is common to all equations. If all regressors are in fact predetermined, then 3SLS reduces to seemingly unrelated regressions (SUR). Thus it may also be seen as a combination of two-stage least squares (2SLS) with SUR.

Applications in social science
Across fields and disciplines simultaneous equation models are applied to various observational phenomena. These equations are applied when phenomena are assumed to be reciprocally causal. The classic example is supply and demand in economics. In other disciplines there are examples such as candidate evaluations and party identification or public opinion and social policy in political science; road investment and travel demand in geography; and educational attainment and parenthood entry in sociology or demography. The simultaneous equation model requires a theory of reciprocal causality that includes special features if the causal effects are to be estimated as simultaneous feedback as opposed to one-sided 'blocks' of an equation where a researcher is interested in the causal effect of X on Y while holding the causal effect of Y on X constant, or when the researcher knows the exact amount of time it takes for each causal effect to take place, i.e., the length of the causal lags. Instead of lagged effects, simultaneous feedback means estimating the simultaneous and perpetual impact of X and Y on each other. This requires a theory that causal effects are simultaneous in time, or so complex that they appear to behave simultaneously; a common example are the moods of roommates. To estimate simultaneous feedback models a theory of equilibrium is also necessary – that X and Y are in relatively steady states or are part of a system (society, market, classroom) that is in a relatively stable state.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Simultaneous equations model


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Lake Como

Lake Como (Lago di Como, ), also known as Lario, is a lake of glacial origin in Lombardy, Italy.

It has an area of 146 km2, making it the third-largest lake in Italy, after Lake Garda and Lake Maggiore. At over 400 m deep, it is the fifth-deepest lake in Europe and the deepest outside Norway; the bottom of the lake is 227 m below sea level. One notable characteristic is its distinctive "Y" shape.

Lake Como has been a popular retreat for aristocrats and wealthy people since Roman times, and a very popular tourist attraction with many artistic and cultural gems. It has many villas and palaces such as Villa Olmo, Villa Serbelloni and Villa Carlotta. Many famous people have owned homes on the shores of Lake Como, including George Clooney, Madonna and Donatella Versace.

In 2014, The Huffington Post described it as the most beautiful lake in the world for its microclimate and environment with prestigious villas and villages.

Name
The lake's official name is Lario (derived from the Latin Larius), but it is rarely used informally, while it is still used in formal language; it is also found in the toponym of some villages along the lake such as Pognana Lario and Mandello del Lario; Italians usually call it Lago di Como.

Geography
The lake is shaped much like an inverted letter "Y". The northern branch begins at the town of Colico, while the towns of Como and Lecco sit at the ends of the southwestern and southeastern branches respectively. The small towns of Bellagio, Menaggio and Varenna are situated at the intersection of the three branches of the lake: a boat service operates a triangular route between them.

Lake Como is fed primarily by the Adda, which enters the lake near Colico and flows out at Lecco. This geological conformation makes the southwestern branch a dead end, and so Como, unlike Lecco, is sometimes flooded.

The mountainous pre-alpine territory between the two southern arms of the lake (between Como, Bellagio and Lecco) is known as the Larian Triangle, or Triangolo lariano. The source of the river Lambro is here. At the centre of the triangle, the town of Canzo is the seat of the Comunità Montana del Triangolo lariano, an association of the 31 municipalities that represent the 71,000 inhabitants of the area.

Climate
According to the Köppen climate classification, the Lake Como area is categorized as humid subtropical climate; average temperatures range from 4 C in January to 25 C in July. Water temperatures reach an average of 26 C in July.

 * Winter – The lake helps to maintain a mild average temperature (7 C) in the surrounding region; snowfall is erratic and occurs mainly at higher elevations; from time to time, periods of frost from the Siberian Anticyclone have been observed.
 * Spring and Autumn – These seasons are well marked and pleasant, with average daytime temperatures around 21 C, and are distinguished by increased precipitation.
 * Summer – During this season average daytime temperatures hover above 27 C and heatwaves have been recurring in recent years with temperatures reaching as high as 38 C, so summer can be quite hot and muggy. This season is also subject to thunderstorms and, at times, violent hailstorms.

Climate change
Until the late 20th century winters used to be quite snowy and cold, with average daily temperatures well below freezing in January and February; however, due to global warming, average temperatures in winter have gradually risen since the turn of the 21st century, reaching a record high of 21 C on January 27, 2024; until the late 2000s summers as well used to be distinguished by pleasant climate but since the 2010s average temperatures have been constantly rising and once rare severe weather events, such as rainstorms and hailstorms, have been happening with increased frequency.

History
At the beginning of the first millennium B.C. during the Iron Age, the Comum oppidum was born and the civilization of Como developed, inserted in the broader Golasecca culture. In 196 B.C. the army of the consul Claudius Marcellus defeats the Celts tribe of the Comenses and conquered the city. Comum was then strengthened and rebuilt after a raid by Rhaetian and repopulated with 3,000 settlers in 77 B.C. Finally, after having reclaimed the marshy area, in 59 B.C. it was re-founded with the name of Novum Comun in its current location on the lake shore at the behest of Gaius Julius Caesar. Pliny the Younger, in one of his Epistulae, describes the lake and its surrounding area as providing plentiful opportunities for fishing and hunting. According to the Notitia Dignitatum, at least since the 4th century, a Praefectus commanding a Roman military fleet was present on the lake.

During the Middle Ages and the Renaissance, the waters of the lake were the scene of military clashes, such as in the 12th century during the war of Milan against Como, which saw the Como fleet in action against the ships of the Milanese and their allies or between 1525 and 1532 due to the Musso war unleashed by Gian Giacomo Medici.

On 28 April 1945, deposed Italian dictator Benito Mussolini was executed in the lakeside village of Giulino, about 180 m from the waterfront.

Tourism
As a tourist destination, Lake Como is popular for its landscapes, wildlife and spas. It is a venue for sailing, windsurfing and kitesurfing.

Although generally considered safe, bathers aiming to find relief from the heat and swimming enthusiasts alike should exercise caution, as a prevailing regulation prohibits diving and swimming both in the city of Como and in the various small villages along the lake. Exceptions are found only in privately managed lidos or designated public beaches where explicit signage permits swimming activities. This prohibition stems from the danger posed by the lake's waters that swiftly transition from shallow to deep near the shoreline and from unpredictable aquatic conditions, which have led to numerous incidents, including drowning cases attributed to sudden thermal shock.

Lakeside villas
The lake is well known for the attractive villas that have been built there since Roman times when Pliny the Younger built the Comedia and the Tragedia resorts. Many villas on the lake shores have admirable gardens that benefit from the mild climate induced by the stabilizing presence of 22.5 km3 of lake water and can sustain many subtropical and Mediterranean plants.

Villa Carlotta was built for the Milanese Marquis Giorgio Clerici in 1690 and occupies a site of over 7 ha at Tremezzo, facing the Bellagio peninsula. An Italian garden (with steps, fountains and sculptures) was laid out at the same time. The villa was later sold to powerful banker and Napoleonic politician Giovanni Battista Sommariva. Stendhal was his guest in 1818, and his visit is recalled at the start of La Chartreuse de Parme. In 1843 it was purchased by Princess Marianne of Nassau as a wedding present for her daughter Carlotta, after whom the villa is now named. The latter, together with her husband Georg II of Saxen-Meiningen, laid out the woodland landscape park in Romantic style. The villa today includes a museum of agricultural implements as well as important works of sculpture by Sommariva's friend Antonio Canova and by Luigi Acquisti.

Villa d'Este, in Cernobbio, was built in 1568 by Cardinal Tolomeo Gallio, a native of the town. In 1816–1817 the villa was home to Caroline of Brunswick, estranged wife of the Prince of Wales and shortly to become queen consort of King George IV of the United Kingdom. The landscaped gardens in the English style are a product of this period. Later in the century, it was turned into a luxury hotel. Today the Villa d'Este is known for attracting celebrity guests.

Villa del Balbianello, famous for its elaborate terraced gardens, lies on a promontory of the western shore of the lake near Isola Comacina. Built in 1787 on the site of a Franciscan monastery, it was the home of explorer Guido Monzino and today houses a museum devoted to his work.

Villa Melzi d'Eril in Bellagio was built in neo-classical style by architect Giocondo Albertolli in 1808–1810 as the summer residence of Duke Francesco Melzi d'Eril, who was vice-president of the Napoleonic Italian Republic. The park includes an orangery, a private chapel, fine statues and a Japanese garden, and is planted, as often on Lake Como, with huge rhododendrons. Nineteenth-century guests at the Villa included Stendhal and Franz Liszt.

Villa Serbelloni, also in Bellagio, hosts the Rockefeller Foundation Bellagio Center, a 50 acre international conference centre set up and managed by the Rockefeller Foundation since 1959, which also operates a "scholar-in-residence" program for scholars from around the world. This is believed to have been the site of Pliny the Younger's villa "Tragedia". Its well-known park was created at the end of the 18th century by Alessandro Serbelloni.

Many famous people have or have had homes on the shores of Lake Como, such as George Clooney.

Transportation
Lake Como is served by a public transport system connecting the various villages on the lake. A motorized service began in 1826 when a steamship with sails, the Lario, was launched by the newly established Società privilegiata per l'impresa dei battelli a vapore nel Regno Lombardo Veneto. Since 1952 the fleet has been managed by a government organisation named Gestione Commissariale Governativa and later Gestione Governativa Navigazione Laghi, which is also responsible for transport services on Lake Maggiore and Lake Garda.

There exist three primary services:

 * Motorship services along the western branch and north towards Colico and back to Como, with additional shuttles to the mid-lake area;
 * Fast services that broadly follow the same route but with fewer stops; the service, which is more expensive, is operated by hydrofoils;
 * Ferries able to carry passengers and cars across the popular tourist destinations Menaggio, Bellagio,Varenna and Cadenabbia.

Sacro Monte di Ossuccio
The Sacro Monte di Ossuccio ("Holy Mount of Ossuccio") is a sanctuary located on a hillside slope between olive groves and woods along the western edge of Lake Como facing Isola Comacina. Fifteen Baroque-inspiredchapels, built between 1635 and 1710, and dedicated to the Mysteries of the Rosary are dotted along the way that leads to the Monastery. This building is the last in the chain and is dedicated to the Coronation of the Virgin.

In 2003, the Sacri Monti of Piedmont and Lombardy, including that of Ossuccio, were added by UNESCO to the World Heritage List.

Economy
Lake Como attracts visitors from around the world and as a consequence the economy of the towns surrounding Lake Como is predominantly dependent on tourism and related activities. The tourism sector stimulates local businesses, including hospitality, restaurants and retail, while also fostering the growth of ancillary services such as transportation, cultural tours and recreational activities. This reliance on tourism has led to significant investments in infrastructure and amenities to accommodate and enhance the visitor experience, making it a vital component of the regional economy.

In literature and the arts

 * Letitia Elizabeth Landon's poem The Lake of Como was published in Fisher's Drawing Room Scrap Book, in 1837. It illustrates a painting by Samuel Prout, engraved by William Miller.

 * In 1818 Percy Bysshe Shelley wrote to Thomas Love Peacock: "This lake exceeds anything I ever beheld in beauty, except the arbutus islands of Killarney. It is long and narrow, and has the appearance of a mighty river winding among the mountains and the forests."

title: Lake Como


text:
Slag

Slag is a by-product of smelting (pyrometallurgical) ores and recycled metals. Slag is mainly a mixture of metal oxides and silicon dioxide. Broadly, it can be classified as ferrous (by-products of processing iron and steel), ferroalloy (a by-product of ferroalloy production) or non-ferrous/base metals (by-products of recovering non-ferrous materials like copper, nickel, zinc and phosphorus). Within these general categories, slags can be further categorized by their precursor and processing conditions (e.g., blast furnace slags, air-cooled blast furnace slag, basic oxygen furnace slag, and electric arc furnace slag). "Slag generated from the EAF process can contain toxic metals, which can be hazardous to human and environmental health".

Due to the large demand for these materials, slag production has also significantly increased throughout the years despite recycling (most notably in the iron and steelmaking industries) and upcycling efforts. The World Steel Association (WSA) estimates that 600 kg of by-products (about 90 wt% is slags) are generated per tonne of steel produced.

Composition
Slag is usually a mixture of metal oxides and silicon dioxide. However, slags can contain metal sulfides and elemental metals.

The major components of these slags include the oxides of calcium, magnesium, silicon, iron, and aluminium, with lesser amounts of manganese, phosphorus, and others depending on the specifics of the raw materials used. Furthermore, slag can be classified based on the abundance of iron among other major components.

Ore smelting
In nature, iron, copper, lead, nickel, and other metals are found in impure states called ores, often oxidized and mixed in with silicates of other metals. During smelting, when the ore is exposed to high temperatures, these impurities are separated from the molten metal and can be removed. Slag is the collection of compounds that are removed. In many smelting processes, oxides are introduced to control the slag chemistry, assisting in the removal of impurities and protecting the furnace refractory lining from excessive wear. In this case, the slag is termed synthetic. A good example is steelmaking slag: quicklime (CaO) and magnesite (MgCO3) are introduced for refractory protection, neutralizing the alumina and silica separated from the metal, and assisting in the removal of sulfur and phosphorus from the steel.

As a co-product of steelmaking, slag is typically produced either through the blast furnace – oxygen converter route or the electric arc furnace – ladle furnace route. To flux the silica produced during steelmaking, limestone and/or dolomite are added, as well as other types of slag conditioners such as calcium aluminate or fluorspar.

Classifications
There are three types of slag: ferrous, ferroalloy, non-ferrous slags, which are produced through different smelting processes.

Ferrous slag
Ferrous slags are produced in different stages of the iron and steelmaking processes resulting in varying physiochemical properties. Additionally, the rate of cooling of the slag material affects its degree of crystallinity further diversifying its range of properties. For example, slow cooled blast furnace slags (or air-cooled slags) tend have more crystalline phases than quenched blast furnace slags (ground granulated blast furnace slags) making it denser and better suited as an aggregate. It may also have higher free calcium oxide and magnesium oxide content, which are often converted to its hydrated forms if excessive volume expansions are not desired. On the other hand, water quenched blast furnace slags have greater amorphous phases giving it latent hydraulic properties (as discovered by Emil Langen in 1862) similar to Portland cement.

During the process of smelting iron, ferrous slag is created, but dominated by calcium and silicon compositions. Through this process, ferrous slag can be broken down into blast furnace slag (produced from iron oxides of molten iron), then steel slag (forms when steel scrap and molten iron combined). The major phases of ferrous slag contain calcium-rich olivine-group silicates and melilite-group silicates.

Slag from steel mills in ferrous smelting is designed to minimize iron loss, which gives out the significant amount of iron, following by oxides of calcium, silicon, magnesium, and aluminium. As the slag is cooled down by water, several chemical reactions from a temperature of around 2600 F (such as oxidization) take place within the slag.

Based on a case study at the Hopewell National Historical Site in Berks and Chester counties, Pennsylvania, USA, ferrous slag usually contains lower concentration of various types of trace elements than non-ferrous slag. However, some of them, such as arsenic (As), iron, and manganese, can accumulate in groundwater and surface water to levels that can exceed environmental guidelines.

Non-ferrous slag
Non-ferrous slag is produced from non-ferrous metals of natural ores. Non-ferrous slag can be characterized into copper, lead, and zinc slags due to the ores' compositions, and they have more potential to impact the environment negatively than ferrous slag. The smelting of copper, lead and bauxite in non-ferrous smelting, for instance, is designed to remove the iron and silica that often occurs with those ores, and separates them as iron-silicate-based slags.

Copper slag, the waste product of smelting copper ores, was studied in an abandoned Penn Mine in California, USA. For six to eight months per year, this region is flooded and becomes a reservoir for drinking water and irrigation. Samples collected from the reservoir showed the higher concentration of cadmium (Cd) and lead (Pb) that exceeded regulatory guidelines.

Applications
Slags can serve other purposes, such as assisting in the temperature control of the smelting, and minimizing any re-oxidation of the final liquid metal product before the molten metal is removed from the furnace and used to make solid metal. In some smelting processes, such as ilmenite smelting to produce titanium dioxide, the slag can be the valuable product.

Ancient uses
During the Bronze Age of the Mediterranean area there were a vast number of differential metallurgical processes in use. A slag by-product of such workings was a colorful, glassy material found on the surfaces of slag from ancient copper foundries. It was primarily blue or green and was formerly chipped away and melted down to make glassware products and jewelry. It was also ground into powder to add to glazes for use in ceramics. Some of the earliest such uses for the by-products of slag have been found in ancient Egypt.

Historically, the re-smelting of iron ore slag was common practice, as improved smelting techniques permitted greater iron yields—in some cases exceeding that which was originally achieved. During the early 20th century, iron ore slag was also ground to a powder and used to make agate glass, also known as slag glass.

Construction
Use of slags in the construction industry dates back to the 1800s, where blast furnace slags were used to build roads and railroad ballast. During this time, it was also used as an aggregate and had begun being integrated into the cement industry as a geopolymer.

Today, ground granulated blast furnace slags are used in combination with Portland cement to create "slag cement". Granulated blast furnace slags react with portlandite (Ca(OH)2), which is formed during cement hydration, via the pozzolanic reaction to produce cementitious properties that primarily contribute to the later strength gain of concrete. This leads to concrete with reduced permeability and better durability. Careful consideration of the slag type used is required, as the high calcium oxide and magnesium oxide content can lead to excessive volume expansion and cracking in concrete.

These hydraulic properties have also been used for soil stabilization in roads and railroad constructions.

Granulated blast furnace slag is used in the manufacture of high-performance concretes, especially those used in the construction of bridges and coastal features, where its low permeability and greater resistance to chlorides and sulfates can help to reduce corrosive action and deterioration of the structure.

Slag can also be used to create fibers used as an insulation material called slag wool.

Slag is also used as aggregate in asphalt concrete for paving roads. A 2022 study in Finland found that road surfaces containing ferrochrome slag release a highly abrasive dust that has caused car parts to wear at significantly greater than normal rates.

Wastewater treatment and agriculture
Dissolution of slags generate alkalinity that can be used to precipitate out metals, sulfates, and excess nutrients (nitrogen and phosphorus) in wastewater treatment. Similarly, ferrous slags have been used as soil conditioners to re-balance soil pH and fertilizers as sources of calcium and magnesium.

Because of the slowly released phosphate content in phosphorus-containing slag, and because of its liming effect, it is valued as fertilizer in gardens and farms in steel making areas. However, the most important application is construction.

Emerging applications
Slags have one of the highest carbonation potential among the industrial alkaline waste due their high calcium oxide and magnesium oxide content, inspiring further studies to test its feasibility in CO2 capture and storage (CCS) methods (e.g., direct aqueous sequestration, dry gas-solid carbonation among others). Across these CCS methods, slags can be transformed into precipitated calcium carbonates to be used in the plastic, and concrete industries and leached for metals to be used in the electronic industries.

However, high physical and chemical variability across different types of slags results in performance and yield inconsistencies. Moreover, stoichiometric-based calculation of the carbonation potential can lead to overestimation that can further obfuscate the material's true potential. To this end, some have proposed performing a series of experiments testing the reactivity of a specific slag material (i.e., dissolution) or using the topological constraint theory (TCT) to account for its complex chemical network.

Health and environmental impact
Slags are transported along with slag tailings to "slag dumps", where they are exposed to weathering, with the possibility of leaching of toxic elements and hyperalkaline runoffs into the soil and water, endangering the local ecological communities. Leaching concerns are typically around non-ferrous or base metal slags, which tend to have higher concentrations of toxic elements. However, ferrous and ferroalloy slags may also have them, which raises concerns about highly weathered slag dumps and upcycled materials.

Dissolution of slags can produce highly alkaline groundwater with pH values above 12. The calcium silicates (CaSiO4) in slags react with water to produce calcium hydroxide ions that leads to a higher concentration of hydroxide (OH-) in ground water. This alkalinity promotes the mineralization of dissolved (from the atmosphere) to produce calcite (CaCO3), which can accumulate to as thick as 20 cm. This can also lead to the dissolution of other metals in slag, such as iron (Fe), manganese (Mn), nickel (Ni), and molybdenum (Mo), which become insoluble in water and mobile as particulate matter. The most effective method to detoxify alkaline ground water discharge is air sparging.

Fine slags and slag dusts generated from milling slags to be recycled into the smelting process or upcycled in a different industry (e.g. construction) can be carried by the wind, affecting a larger ecosystem. It can be ingested and inhaled, posing a direct health risk to the communities near the plants, mines, disposal sites, etc.

title: Slag


text:
Talk:Lake Como

Name
About the name in English: I find that "Lake of Como" is the most accurate, and it is the literally translation of the Italian name "Lago di Como". Looking for it on Google I also found "Como Lake", but it misleads into thinking that "Como" might be the name of the lake itself, while it's not, it's just the main town on the lake's shores.

And its real name "Lario" is used very rarely, so I don't think it should be used, I'm placing a redirect for it now. Laz 23:19 20 Jun 2003 (UTC)

 * I know it isn't a perfectly accurate translation, but in English, the lake is known as "Lake Como" (59,400 Google hits on English-language pages) not "Lake of Como" (1,960 hits on English-language pages). As this is the English-language Wikipedia, the convention is to give thing the most common name used in an English-language context. Therefore, I'm moving this to Lake Como. --Camembert

 * As you wish, but I'm still deeply convinced that this is a big mistake and it misleads people into thinking that Como is the name of the lake, while it is not at all. Google hits give the most common name, and this generally works for our purposes; but if the most common name is wrong, IMHO a good Encyclopedia should point it out and list the correct name. But I'm sure not going to argue anymore about this, we're not here to argue. I'll leave it up to someone else. Laz 12:40 21 Jun 2003 (UTC)

 * If you want to explain details about the lake's name, pointing out that in Italian the lake isn't called "Como" as such, then great - do so in the article. But the title should be the most common name used in an English-language context (see Naming conventions (common names)). Incidentally, the Columbia, Macmillan and Hutchinson encyclopaedias all call this "Lake Como", as does the Penguin Encyclopaedia of Places. --Camembert

 * I was thinking the same thing actually... a note similar to Kings Cross station's one. Ok, thanks for the tip. Laz 13:08 21 Jun 2003 (UTC)

When I saw "Lake of Como" I said "Whaaa!? Who let the Italians run amuck!" :-) It may be a fine literal translation, but "Lake of X" is extremely uncommon, Lake of the Woods is the only one that comes to mind. To borrow a line from books on software localization, one should give precedence to native speakers' ideas about what sounds right. Stan 05:49, 11 May 2004 (UTC)

 * Ok, I moved Lago di Como back to Lake Como. -- User:Docu

"From running a search on Google one might conclude that "Lake Como" is much more common, in English, than "Lake of Como"." - I removed this note because Google results are rather ephmeral and not necessarily a reliable source either. People who are that interested should come read this talk page. Stan 16:32, 11 Jun 2004 (UTC)

Sincerly I think is much more accurate to name the article "Lario": Italians care about history and it's the original historic name. I find the use of Lake of something profoundly wrong and too much simplified. For example King Louis IV is offently called The King Sun... but fortunately Wikipedia uses his common name... why not do the same with our lakes? - Quattrop 17:20 03 Aug 2005 (SGT)

Lario isn't so rarely as you think, here we often use to say Lario, and citizens of Como are comaschi or lariani --2001:B07:645B:A5C0:2E2B:F9FF:FEB2:16A0 (talk) 18:07, 28 March 2021 (UTC)

Risotto book link
Although the book is related to the Villa d’Este, I didn’t find this link to Amazon particulary informative I’ve removed it rom the article for now—Ian Spackman 23:38, 16 August 2006 (UTC)
 * Tales of Risotto: Culinary Adventures from Villa d'Este by Jean Govoni Salvadore and Luciano Parolari.
 * addition of the book was probably link spam. See User_talk:66.108.157.73 --Mattarata 22:58, 24 August 2006 (UTC)

Images
This article includes superb photos. If you took them, uploaded them, added them, or captioned them, please accept my thanks. —Ian Spackman 14:37, 11 November 2006 (UTC)

Speaking of images, does the article really need 3 panoramas? Splamo 22:04, 12 November 2006 (UTC)

Fish ?
Are there fish in the lake? —The preceding unsigned comment was added by 209.206.165.24 (talk) 10:48, 28 January 2007 (UTC). Yes, a lot --2001:B07:645B:A5C0:2E2B:F9FF:FEB2:16A0 (talk) 18:09, 28 March 2021 (UTC)

Lake Como
All municipality articles note which region and province they are located in, how far they are from Milan, etc, but only a few describe their location on Lake Como! -- User:Docu
 * Yes. Most of those little articles were created by a robot which knows about regions and provinces and demography and so forth, and can work out distances and directions from the coordinates, but knows nothing about lakes.

 * if you wanted to work through them—or even some of them—, adding the information, that would be great. Cheers! —Ian Spackman 13:12, 1 April 2007 (UTC)

 * Indeed. I will try to do one or the other. I already went through those on Lake Lugano. -- User:Docu

External links moved here from the article
I am moving all of them (except commonscat here. There are far, far too many and really they ought to be restored only one at a time, after carefully reviewing them for their value. —Ian Spackman 11:08, 16 May 2007 (UTC)
 * A tourism portal, developed by a partnership between public bodies and private consortia.
 * A short factual article using the Guida azzura di Como e dintorni (1989) as its principal source.
 * Map of Lake Como
 * City of Como Official site of the Commune
 * City of Lecco Official site of the Commune
 * Provincia di Lecco Official site of the Province
 * Provincia di Como Official site of the Province
 * Lecco APT touristic promotional association
 * Official site of the Villa Carlotta
 * History of the Villa d’Este and its Gardens . Commercial site of the Villa d’Este hotel offering some brief historical coverage.
 * Locanda dell'Isola Comacina. Commercial site of an Isola Comacina pub, with information on the island and its history.
 * http://www.larioonline.it Lake Como travel guide
 * Lago di Como Italian Wikipedia article

Update October 2009:

I have added the dmoz template to aid readers in finding sites related to the subject and have left in place the commons link template. I have left in the site Ferry Services on Italian Lakes - Lake Como  on the basis that it is the official site of the ferry operator, which is discussed in the article itself.

I am not at all convinced however that the other two sites—decent as they are in their different ways—pass the rather stringent tests for inclusion set out in WP:EL: e.g. ‘one should avoid…any site that does not provide a unique resource beyond what the article would contain if it became a Featured article’. I am therefore removing them here and would suggest that they be restored only after consensus has been achieved following discusion on this talk page:

 * Official Virtual Tour - Lake Como
 * Not official in any meaningful sense—though the municipality of Como is one of the sponsers. A collection of decent quality panoramic images displayed using Flash.

 * News and Events - Lake Como
 * The English translation can be a bit slapdash—on the page on Tremezzo, for instance, the Villa Carlotta seems to have become the Villa Carla—but is certainly no worse than many similar sites and in general it is often rather informative. Pages like the one on Como Winds, could be usefully read by someone improving our article.

—Ian Spackman (talk) 13:29, 26 October 2009 (UTC)

Pollution
The cited bit about the lake's pollution had been deleted. I added it back in as the information seemed both relevant and well supported. Let me know if I missed something. —Preceding unsigned comment added by 71.211.119.246 (talk) 20:19, 15 August 2008 (UTC)

Depth of the lake
In the German article, a depth of 425 meters for the lake is mentionend. This is a very new information, in 2007 scientists found out the new depth of the lake. This is the information, I have found on the German wikipedia article of Lake Como. But there are no sources for this information. Does anyone know any about the new depth (if so, wikipedia needs some more information) or is 410 m still correct? thanks, --Holger1974 (talk) 11:34, 21 December 2008 (UTC)

Einstein's Fateful Visit to Lake Como
Perhaps one of the most interesting aspects of Einstein's personal life was the fact that he had an illegitimate daughter whose existence was only known to a few during Einstein's lifetime. In fact the rest of world was unaware that Einstein had a third child until 1986, long after his death.

Einstein impregnated Mileva Maric during a vacation at Lake Como in April of 1901 (See Walter Isaacson's biography pg 63-67). This was an extremely fateful vacation, as it lead to Einstein marrying Maric and the birth of their daughter Lieserl. It is unclear what happened to Lieserl, but most likely she either died or was adopted. — Preceding unsigned comment added by 108.161.215.195 (talk) 06:55, 12 January 2013 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 5 external links on Lake Como. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20091106054721/http://www.lifeinitaly.com/italian-movies/clooney.asp to http://www.lifeinitaly.com/italian-movies/clooney.asp
 * Added archive https://web.archive.org/web/20091015075933/http://www.navigazionelaghi.it/eng/com/storia/como/1.html to http://www.navigazionelaghi.it/eng/com/storia/como/1.html
 * Added archive https://web.archive.org/web/20100505021209/http://www.navigazionelaghi.it/eng/sed/sede.html to http://www.navigazionelaghi.it/eng/sed/sede.html
 * Added archive https://web.archive.org/web/20100316160050/http://www.navigazionelaghi.it/eng/com/nlc_ora.html to http://www.navigazionelaghi.it/eng/com/nlc_ora.html
 * Added archive https://web.archive.org/web/20070503014800/http://www.villacarlotta.it/sito/index2.php to http://www.villacarlotta.it/sito/index2.php

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 17:04, 9 December 2017 (UTC)

Villa Serbelloni internal link weirdnesss
hovering over or clicking the Villa_Serbelloni link takes you to the Rockefeller Foundation link

(i would assume because it's the next valid internal link on the page?)

because there is no English version of that page yet (Italian, Deutsche, and Chinese versions of the Villa_Serbelloni page exist though)

but i also found this subsection about the villa on another page: https://en.wikipedia.org/wiki/Bellagio,_Lombardy#Villa_Serbelloni and that page is in just about every language ...

so.......... should the link point to that subsection for the villa on the Bellagio article instead of the main article only in three languages?

just a thought... was gonna just make the change, but then figured posting the thought here might be better before taking further action

Cheers, Chris Heath (talk) 06:57, 27 June 2019 (UTC)

Flooding
The article currently says: 'Lake Como is fed primarily by the Adda, which enters the lake near Colico and flows out at Lecco. This geological conformation makes the southwestern branch a dead end, and so Como, unlike Lecco, is often flooded.' I'm not sure that is a sufficient explanation for the reportedly frequent flooding of Lecco - if indeed that is a fact, since this is not referenced anyway. Unless under the influence of sustained high winds from the northern quarter, there should be no particular water level difference between different parts of the lake, and I'd expect these to be small differences anyway. thanks Geopersona (talk) 17:25, 21 March 2023 (UTC)

Swimming
As a resident of Como and a major contributor to the city's article (https://xtools.wmcloud.org/authorship/en.wikipedia.org/Como/) I felt compelled to add a note to swimmers to be cautious when deciding to dive and swim in Lake Como - each summer the local media reports 5/10 cases of people drowning after carelessly diving and swimming where they think it is safe; Lake Como instead is quite dangerous, especially for unexperienced and casual swimmers: in many places the waters get quickly very deep (15 meters / 50 feet) a few meters from the shore; the lake's waters can also be very cold, even during hot summer days, and cause a cold shock to careless divers, which has resulted in many deaths, even among healthy young people and athletes. Swimming is also dangerous because of unexpected and sudden underwater currents and whirlpools. Since the page is visited on average 1500 times a day, mostly for touristic info, i hope this note of caution contributes to avoid another preventable death. Itemirus (talk)  06:32, 18 August 2023 (UTC)

title: Talk:Lake Como


text:
Lake Como (disambiguation)

Lake Como may refer to:

Lakes

 * Italy
 * Lake Como, a lake of glacial origin in Lombardy, Italy
 * United States
 * Lake Como (Minnesota), a lake in Saint Paul, Minnesota
 * Lake Como (New York), a lake in Cayuga County, New York
 * Lake Como (Montana), a lake in Ravalli County's Bitterroot Valley
 * Lake Como (Colorado), a lake in Alamosa County, Colorado

Places

 * United States
 * Lake Como, Florida, unincorporated community
 * Lake Como, Mississippi, unincorporated community
 * Lake Como, New Jersey, borough
 * Lake Como, Pennsylvania, unincorporated community
 * Lake Como, Wisconsin, unincorporated community

title: Lake Como (disambiguation)


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Rod Grams

Rodney Dwight Grams (February 4, 1948 – October 8, 2013) was an American politician and television news anchor who served in both the United States House of Representatives and the U.S. Senate. A local news anchor, Grams became well-known for working at Twin Cities station KMSP-TV from 1982 until 1991. He was a member of the Republican Party.

Grams was born on a farm in Princeton, Minnesota, and worked at several other news stations throughout the Midwest and Great Plains before serving as KMSP's senior news anchor. After retiring from television, he launched a successful bid for Congress in Minnesota's 6th congressional district against embattled Democratic incumbent Gerry Sikorski in 1992. He served one term, opting to run for the U.S. Senate seat being vacated by retiring Republican David Durenberger in 1994. He won the 1994 Senate election and was defeated for reelection by Mark Dayton in 2000.

Grams sought election to his old Senate seat in 2006, but dropped out before the Republican primary. He also ran again for election to the House in Minnesota's 8th congressional district in 2006 as the Republican-endorsed candidate but lost to incumbent Democrat Jim Oberstar.

Early life
Grams was born on a farm in Princeton, Minnesota. He attended Brown Institute (1966–1968), Anoka-Ramsey Community College (1970–1972), and Carroll College (1974–1975).

Grams spent 23 years in television and radio broadcasting before launching a career in politics. From 1982 to 1991 he was the senior news anchor at KMSP-TV in Minneapolis/St. Paul. Before that, he worked as a news anchor/producer for KFBB-TV in Great Falls, Montana; WSAU-TV in Wausau, Wisconsin; and WIFR-TV in Rockford, Illinois. Before entering broadcasting, Grams worked at an engineering consulting firm for seven years. In 1985, he formed Sun Ridge Builders, a Twin Cities construction and residential development company, serving as its president and CEO. He was involved in architectural design and the use of solar energy in residential homes.

Early political career
Grams launched his political career by winning the 1992 Republican nomination in Minnesota's 6th congressional district. He defeated Democratic incumbent Gerry Sikorski in the general election. During the campaign, Grams benefited from high name recognition in the district—from years as news anchor at KMSP-TV—and Sikorski's involvement in the House banking scandal. He served in the 103rd Congress as a member of the House of Representatives, and in the 104th, 105th, and 106th congresses as a member of the U.S. Senate.

U.S. Senate
After David Durenberger announced he would not seek reelection, Grams surprised many by announcing, just months into his first term in the U.S. House, that he would run for the Senate. Grams faced opposition for the Republican endorsement from State Senator Gen Olson, Bert McKasy (Durenberger's former chief of staff), and Doug McFarland. During the party endorsement process, the Grams campaign closely aligned itself with supporters of Allen Quist, who was challenging incumbent Governor Arne Carlson for reelection in the 1994 gubernatorial race. Weeks before the state Republican convention, McFarland dropped out of the Senate race after endorsing Quist for governor and joining his ticket as Quist's running mate.

After numerous ballots at the convention in St. Paul, Grams won the Republican endorsement over Olson and McKasy. He also won the Republican primary against Arne Carlson's Lieutenant Governor Joanell Dyrstad, who had been replaced as his running mate by State Senator Joanne Benson. In the general election against Democratic Farmer Labor candidate Ann Wynia and Independence Party candidate Dean Barkley, Grams won, 49% to 44%.

Grams ran for reelection in 2000, losing to Mark Dayton. During the campaign, Grams's wife Christine Gunhus was revealed to have written anonymous disparaging emails about Grams's potential Democratic rival Mike Ciresi from her home computer. She received a fine and suspended sentence for violating political advertising regulations. The Grams campaign also ran a commercial during the campaign featuring Grams's mother, Audrey Grams, that ended with her saying, "Uffda, vote for Rod."

Post-Senate career
After his 2000 reelection defeat, Grams went back into private business and in 2004 became the owner of three radio stations in Little Falls, Minnesota. He attempted a political comeback in the 2006 U.S. Senate campaign. He sought the GOP nomination for his former Senate seat, facing Mark Kennedy and Gil Gutknecht. After a poor showing early in the endorsement process, Grams dropped his Senate candidacy and ran in the 2006 U.S. House election, challenging the incumbent Jim Oberstar in Minnesota's 8th congressional district. Oberstar defeated Grams handily.

Grams remained active in politics and interested in running for public office. In 2008, he considered challenging incumbent Norm Coleman for the Republican U.S. Senate nomination but was too busy in his private life to make a run, saying, "And my wife (Chris) would have killed me if I would have, because of some things that we're doing." It also became unclear whether Grams would run as a Republican. In an interview, he expressed disappointment at the perceived failings of the party, going as far as to wonder whether he could still call himself a Republican or vote for Republican candidates.

Grams considered a 2010 run for governor of Minnesota, saying, "I'm so damn unhappy with the Republicans right now ... I'm so unhappy with the candidates that we have I could puke. I wanted to get out there and mix it up." But Grams endorsed Republican Tom Emmer in the 2010 campaign for governor. Emmer lost the election to Mark Dayton.

Illness and death
On September 4, 2013, it was announced that Grams had been battling colon cancer since 2012 and was receiving hospice care at his home. He died on October 8, 2013, aged 65, at his home in Crown, Minnesota.

title: Rod Grams


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The Odyssey: A Modern Sequel

The Odyssey: A Modern Sequel is an epic poem by Greek poet and philosopher Nikos Kazantzakis, based on Homer's Odyssey. It is divided into twenty-four rhapsodies as is the original Odyssey and consists of 33,333 17-syllable verses. Kazantzakis began working on it in 1924 after he returned to Crete from Germany. Before finally publishing the poem in 1938 he had drafted seven different versions. Kazantzakis considered this his most important work. It was fully translated into English in 1958 by Kimon Friar.

Synopsis
Odysseus (Ulysses) returns to Ithaca and decides to undertake new adventures after he quickly becomes dissatisfied with his quiet family life and they too with his brutality. First he travels to Sparta to save Helen, the wife of the king of Sparta Menelaus, whose abduction by Paris had led to the Trojan War. He goes to Crete where a conspiracy dethrones the king. He is abandoned by Helen who runs off with a blonde barbarian and continues to Egypt, where again a workers' uprising takes place. He leaves again on a journey up the Nile eventually stopping at the lake-source. Upon arrival his companions set up camp and he climbs the mountain in order to concentrate on his god. Upon his return to the lake he sets up his city based on the commandments of his religion. The city is soon destroyed by an earthquake. Odysseus laments his failure to understand the true meaning of god with the sacrifice of his companions. His life transforms into that of an ascetic. Odysseus meets Motherth (an incarnation of the Buddha), Kapetán Énas (English: Captain Sole, literally "Captain One", a Greek folk expression for people who are insubordinate and single-minded to a fault), alias Don Quixote, and an African village fisherman, alias Jesus. He travels further south in Africa while constantly spreading his religion and fighting the advances of death. Eventually he travels to Antarctica and lives with villagers for a year until an iceberg kills him. His death is glorious as it marks his rebirth and unification with the world.

Themes
The Odyssey represents Kazantzakis' ideology and metaphysical concerns. A central theme is the importance of struggle for its own sake, as opposed to reaching a final goal.

title: The Odyssey: A Modern Sequel


text:
User:Dino

Please click color="#5a3696">here to leave me a message on my Talk page. Thanks.

Agree to multi-license all my contributions, with the exception of my user pages, as described below:

Why write for wikipedia? Believe in open content / open source. It keeps my brain active. Write of things which know, of a few things don't know and probably shouldn't touch, and am forced to research things and learn about them, if am to write a decent article on a topic.

Plus, it's kinda fun. As of December 2, 2004, I am a Wikipedia administrator. Am so busy with other distractions scarce write anything. All my edits,, contributions, here

Zip me an e-mail for any questions or concerns, or use my Talk page — check it with relative frequency.

I am trained in mathematics (primarily) & chemistry & live in Boulder, Colorado, USA.

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title: User:Dino


text:
David Durenberger

David Ferdinand Durenberger (August 19, 1934 – January 31, 2023) was an American politician and attorney. Durenberger represented Minnesota in the United States Senate as a Republican from 1978 to 1995. He left the Republican Party in 2005 and became a critic of it, endorsing Democratic presidential nominees Hillary Clinton and Joe Biden in 2016 and 2020, respectively.

Early life
Durenberger was born in St. Cloud, Minnesota, the son of Isabelle Marie (née Cebulla) and George Gephard Durenberger. He was a Roman Catholic of German and Polish descent. His father was the athletic director and a coach at Saint John's University in Collegeville, Minnesota, and the family lived on campus.

Durenberger graduated from St. John's Prep School there in 1951, and from the university in 1955. He attended the University of Minnesota Law School and earned his Juris Doctor in 1959. At St. John's he was the top-rated cadet in his ROTC class, and after college was a lieutenant in the U.S. Army Counter-Intelligence Corps in 1956 and a captain in the United States Army Reserve from 1957 to 1963.

Professional career
After law school, Durenberger was employed by a South St. Paul law firm with strong political connections. It had been founded in 1929 by Republican Harold Stassen, later the governor of Minnesota from 1939 to 1943, and Elmer Ryan, a Democrat who was member of the U.S. House of Representatives from 1935 to 1941. When Durenberger joined it was headed by Harold LeVander. The firm took the name LeVander, Gillen, Miller and Durenberger.

LeVander, a Republican, was elected governor of Minnesota in 1966 and took office in January 1967, and Durenberger became his executive secretary from then until the end of LeVander's term in 1971. He then joined the H.B. Fuller Company as in-house counsel, corporate secretary, and manager of international licensing until 1978. He also served as chair of the Metropolitan Open Space Advisory Board from 1972 to 1974 and was on the Minnesota State Ethical Practices Board from 1974 to 1978.

United States Senate
On November 7, 1978, Durenberger was elected to the United States Senate in a special election to complete the unexpired term of Senator Hubert Humphrey, who died earlier in the year; Humphrey's wife Muriel held the seat until Durenberger's election. Durenberger was reelected in 1982 and again in 1988, defeating Mark Dayton and Minnesota Attorney General Skip Humphrey, respectively.

In the 99th Congress, Durenberger chaired the Select Committee on Intelligence and the Health Subcommittee of the Senate Finance Committee, giving him a leadership role in national health reform. He also chaired the Intergovernmental Relations Subcommittee, led President Ronald Reagan's New Federalism effort in 1982, and was a 14-year member of the Advisory Committee on Intergovernmental Relations. He was a member of the Senate Environment Committee, the Government Affairs Committee, and the committee now known as the Health, Education, Labor, and Pensions Committee, and served as vice chair of the Pepper Commission in 1989–90.

Durenberger was Senate sponsor of the Medicare Catastrophic act, the AHCPR (now AHRQ) on voting rights for the disabled, the Americans with Disabilities Act, President George H. W. Bush's 1000 Points of Light, President Bill Clinton's National and Community Service Act, National Service Learning, the Consumer Choice Education Act, the Safe Drinking Water Act, the Direct Lending Act, and the Women's Economic Equity Act. Durenberger voted for the bill establishing Martin Luther King Jr. Day as a federal holiday and the Civil Rights Restoration Act of 1987 (as well as to override Reagan's veto). He voted to confirm Robert Bork and Clarence Thomas to the U.S. Supreme Court.

Misuse of public funds
In 1990, the senate voted 96–0 to censure Durenberger for ethics violations related to evading limits on $100,000 in speaking fees and using his condominium in Minneapolis to collect $40,000 in travel reimbursements. The Minnesota Supreme Court indefinitely suspended Durenberger's Minnesota law license on January 11, 1991, pursuant to a stipulation. It reinstated his license on March 22, 2000.

Durenberger did not run for reelection in 1994 and was succeeded by Rod Grams. In 1995, he pleaded guilty to charges of misuse of public funds while in office and was sentenced to one year of probation.

Post-Senate life
In a 2005 interview, Durenberger said he no longer supported the Republican Party but did not support the Democratic Party either. He also said that Democrats are better equipped to handle health care and that President George W. Bush was wrong about the Iraq War. In 2010, Durenberger endorsed his former chief of staff, Independence Party member Tom Horner, for governor.

Durenberger chaired the National Institute of Health Policy (NIHP) and was a Senior Health Policy Fellow at the University of St. Thomas in Saint Paul. He served on the board of National Coalition on HealthCare. He has also served on national health commissions and boards, including the Medicare Payment Advisory Commission and Board of the National Commission on Quality Assurance (NCQA), and the Kaiser Commission on Medicaid and the Uninsured.

Durenberger endorsed former Secretary of State Hillary Clinton for president in 2016 and endorsed former Vice President Joe Biden for president in 2020. He was a member of the ReFormers Caucus of Issue One.

Personal life
Durenberger's first wife, Judith, died of cancer in 1970. He and his second wife, Penny, separated in 1985. Durenberger married his third wife, Susan, in 1995. He had four sons from his first marriage and two stepchildren.

Durenberger died of heart failure at home in St. Paul on January 31, 2023, at age 88.

Writings
A collection of Durenberger's senatorial files is held by the Minnesota Historical Society. It documents his three terms in the United States Senate and is strongest in its documentation of the third (1989–95). The papers are perhaps most significant for the information they contain about his interest in, and legislative activities regarding, health policy and health care reform issues.

Durenberger's books include When Republicans were Progressive, which traces the history of Minnesota's Republican party from the era of Stassen, a moderate Republican governor who took office in 1939, to the ascent of a more conservative strain within the party in the late 1980s (Durenberger lamented the polarization of more recent politics); Neither Madmen nor Messiahs: A Policy of National Security for America (1984), on defense policy; and Prescription for Change (1986), on health care reform.

title: David Durenberger