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Successive partial derivatives - Applications of Differentiation
Consider the functionu u = f( x, y) .
Successive partial derivatives
Consider the functionu u = f( x, y) . From this we can find are functions of x and y, then they may be differentiated partially again with respect to either of the independent variables, (x or y)
denoted by
These derivatives are called second order partial derivatives. Similarly, we can find the third order partial derivatives, fourth order partial derivatives etc., if they exist. The process of finding
such partial derivatives are called successive partial derivatives.
If we differentiate u = f(x,y) partially with respect to x and again differentiating partially with respect to y, we obtain
Similarly, if we differentiate u = f(x,y) partially with respect to y and again differentiating partially with respect to x,
If u(x,y)is a continuous function of x and y then
Homogeneous functions
A function f(x,y) of two independent variables x and y is said to be homogeneous in x and y of degree n if f (tx , ty ) = t ^n f ( x, y) for t > 0.
Tags : Applications of Differentiation , 11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation : Successive partial derivatives | Applications of Differentiation
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Gann Theory: A Pattern, Price and Time Combined Analysis
By Isfaqur Rahman
Gann Theory: A Pattern, Price and Time Combined Analysis
Gann Theory: A Pattern, Price and Time Combined Analysis
• The study of pattern, price, and time and how their relationships affect the market.
Gann based predictions of price movements on three premises:
• Price, time, and range are the only three factors to consider.
• The markets are cyclical in nature.
• The markets are geometric in design and in function.
Based on these three premises, Gann’s strategies revolved around three general areas of prediction:
• Price study: This uses support and resistance lines, pivot points, and angles.
• Time study: This looks at historically reoccurring dates, derived by natural and social means.
• Pattern study: This looks at market swings using trendlines and reversal patterns.
Gann Angles
• Gann Angles are drawn from major price peaks and bottoms and are used to show trend lines of support and resistance.
• This allows the analyst to forecast where the price is going to be on a particular date in the future.
Constructing Gann Angles
Determine the time units:
• In most cases, the intermediate-term (such as one- to three-month) charts produce the optimal amount of patterns.
Determine the high or low from which to draw the Gann lines:
• The most common way to accomplish it is to use other forms of technical analysis—such as Fibonacci levels or pivot points.
Determine which pattern to use:
• The two most common patterns are the 1×1 (left figure), the 1×2 (right figure), and the 2×1.
• The 1X2 means the angle is moving one unit of price for every two units of time.
• The 1X1 is moving one unit of price with one unit of time.
• The 2X1 moves two units of price with one unit of time.
Draw the patterns:
• The direction would be either downward and to the right from a high point, or upward and to the right from a low point.
Look for repeat patterns further down the chart:
• This technique is based on the premise that markets are cyclical.
Gann Angles Provide Support and Resistance
• Uptrending angles provide support.
• Downtrending angles provide the resistance.
• Because the analyst knows where the angle is on the chart, he or she is able to determine whether to buy on support or sell at the resistance.
Gann Angles Determine Strength and Weakness
• Trading on or slightly above an uptrending/downtrending 1X1 angle means that the market is balanced.
• Trading on or slightly above an uptrending/downtrending 2X1 angle means that the market is in a strong uptrend/downtrend.
• Trading at or near the 1X2 means the trend is not as strong.
• Anything under the 1X1 is in a weak position.
Gann Angles Can be Used for Timing
• The basic concept is to expect a change in direction when the market has reached an equal unit of time and price up or down (1X1 angle).
• This timing indicator works better on longer-term charts, such as monthly or weekly charts; this is because the daily charts often have too many tops, bottoms, and ranges to analyze.
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Strut Like a Sheep
Sheep are herd animals who follow each other without asking questions, so somebody has to tell them which way to walk. So we’re loving this video filmed from an airplane high above, showing a giant
herd of sheep streaming across the grass and making wild flowing shapes almost like water. You can see tiny sheepdogs running around the group to chase them. How many sheep do you guess there are?
That’s a lot of legs, a lot of baahs, and a lot of future sweaters.
Wee ones: Sheep are white. Try to find 4 white things in your room!
Little kids: How many more legs than you does a sheep have? Bonus: In a row of 9 sheep, which number sheep is the very middle one?
Big kids: If there are 400 sheep, and half of them flow through the fence into the next yard, how many more sheep are still behind? Bonus: If each minute after that, half of the sheep who are left
pass through, how many still haven’t passed through the fence after 2 minutes?
The sky’s the limit: If there are 11 times as many sheep as dogs, and there are 600 animals in total, how many of each animal are there?
Wee ones: Items might include bedsheets, pillow cases, socks, sneakers, or sheets of paper.
Little kids: 2 more legs. Bonus: The 5th sheep.
Big kids: 200 sheep. Bonus: 50 sheep.
The sky’s the limit: If there are 11 sheep for each dog, the animals come in sets of 12. There are 50 12s in 600, so there would be 50 dogs and 550 sheep (which is in fact 50 x 11).
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Capital Budgeting Project Analysis Relevant information: The firm uses a 3-year cutoff when using the payback method.The hurdle rate used to evaluate capital budgeting projects is 15%.The cash flows f
Answered You can buy a ready-made answer or pick a professional tutor to order an original one.
Capital Budgeting Project Analysis Relevant information: The firm uses a 3-year cutoff when using the payback method.The hurdle rate used to evaluate capital budgeting projects is 15%.The cash flows
Capital Budgeting Project Analysis
Relevant information:
1. The firm uses a 3-year cutoff when using the payback method.
2. The hurdle rate used to evaluate capital budgeting projects is 15%.
The cash flows for projects A, B and C are provided below.
Project A Project B Project C
Year 0 -30,000 -20,000 -50,000
Year 1 0 4,000 20,000
Year 2 7,000 5,000 20,000
Year 3 20,000 6,000 20,000
Year 4 20,000 7,000 5,000
Year 5 10,000 8,000 5,000
Year 6 5,000 9,000 5,000
1. Assume the projects are independent and answer the following:
□ Calculate the payback period for each project.
□ Which project(s) would you accept based on the payback criterion?
□ Calculate the internal rate of return (IRR) for each project.
□ Which projects would you accept based on the IRR criterion?
□ Calculate the net present value (NPV) for each project.
□ Which projects would you accept based on the NPV criterion?
2. Assume the projects are mutually exclusive and answer the following:
□ Which project(s) would you accept based on the payback criterion?
□ Which projects would you accept based on the IRR criterion?
□ Which projects would you accept based on the NPV criterion?
Submission Instructions:
Submit completed work in a Excel spreadsheet.
Show more
Click here to download attached files: capital budgeting.xlsx
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Export Reviews, Discussions, Author Feedback and Meta-Reviews
Submitted by Assigned_Reviewer_4
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
This paper pursues an intriguing concept that has been proposed and considered by others as well (as appropriately cited in the present paper): given more data, can we reduce the running time
required to obtain a given risk. This is clearly an important question, and while there has been some progress, the surface has really barely been scratched.
This paper considers the problem of sparse regression, in particular in the setting of no noise — the Regularized Linear Inverse Problem. The main idea is to regularize more heavily when there are
more data. It has been observed and also proved rigorously, that regularizing increases the rate of convergence, but more interestingly, may not change the solution (by much, or at all). Two
important references in this respect are papers by Wotao Yin and Ming-Jun Lai, that demonstrate precisely this idea, with the goal of showing a very similar conclusion as that reached in this paper.
In those papers, they show that if the sensing matrix A satisfies some additional properties (e.g., better RIP) then one can regularize and still recover (nearly) the same answer, but without sending
the regularizer to zero (just like what is done in this paper). If one has more data, then it is again fairly straightforward calculation to show that the sensing matrix A (assuming it comes from a
suitable random ensemble) will indeed satisfy RIP with a stronger constant. Then, since the regularizer controls the strong convexity parameter of the problem, this speeds up convergence (and in
particular, guarantees a global linear rate). Another paper that is relevant and has a very similar idea but without using RIP, is by Agarwal, Negahban and Wainwright, where the authors show that
thanks to restricted strong convexity and restricted smoothness, gradient methods have global geometric (linear) convergence. The key connection with the present paper is that convergence time
depends explicitly on the RSC and RSM parameters, and one can show (as the authors there do) that these improve when one has more data.
The algorithm in the present paper is a dual smoothing algorithm. Strong convexity is exploited in order to convert the dual solution to the corresponding (and unique) primal solution.
The organization and writing of this paper is not as clear as it might be. One thing that would improve the delivery of the results, is some simple calculations for a setting where, say, A comes from
Gaussian design. Computing \mu(m) here should not be that difficult, yet would help tell the story. That is, it would be nice to have some calculations analogous to Fact 2.3, but not just for when
exact recovery holds, but rather, how big the regularizer can be while still guaranteeing exact recovery. The current connection to regularizing and Fact 2.3 is not completely clear to me; it should
be, as it is one of the core pieces of the paper.
Q2: Please summarize your review in 1-2 sentences
In general the paper is pursuing a nice direction, for which there has been much ground work laid, but the details are not all as clear as they could be. I’m not sure that they are so in this paper
either, though the authors are certainly pushing in this direction.
Submitted by Assigned_Reviewer_8
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
The paper deals with time-data tradeoffs that appear in noiseless (convex) regularized linear inverse problems (RLIP) (e.g. compressed sensing). The authors show that when an excess data is available
to solve a RLIP, the regularizer can be smoothed and the RLIP can be relaxed to a simpler problem that can be solved more efficiently in terms of computational complexity. To establish this tradeoff
the authors use a standard iterative dual smoothing optimization algorithm and show that in the presence of excess data, minimizing the smoothed regularizer can achieve the same level accuracy with
less iterations. On the other hand the authors explicitly characterize when the smoothed RLIP problem will yield the same (true) solution as the original problem by using the notion of the
statistical dimension that precisely characterizes the phase transition curve of RLIPs with random Gaussian sensing matrices. The authors present their results for the problems of sparse signal
estimation via the l-1 norm, and low rank matrix estimation via the nuclear norm. They also point to other examples and noisy signal estimation in subsequent work.
Quality: The authors use modern convex analysis tools (e.g. statistical dimension of convex cones) to approach the problem of time-data tradeoffs in statistical estimation in a principled and well
structured way.
Clarity: The paper is written in a very clear way and is self-contained.
Originality and significance: The authors cite the innovative work of Chandrasekaran and Jordan as another example of time-data tradeoffs in RLIP. However, they propose a different approach
(smoothing the regularizer instead of enlarging the constraint set) which appears to be more flexible and applicable to problems that appear often in practice.
My only concern is the dependance of the method on the choice of the optimization algorithm. Are these results also applicable to other more specialized algorithms (e.g. approximate message passing
for l-1 minimization) that offer strong performance guarantees?
Q2: Please summarize your review in 1-2 sentences
The paper addresses the problem of time data tradeoffs in regularized linear inverse problems in a novel way, that is theoretically sound, and practically applicable.
Submitted by Assigned_Reviewer_36
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
In regularized linear inverse problems, the sample complexity for recovering the true signal has recently been well understood. This paper agues that excess observations than needed in the sample
complexity bound can be used to reduce the computational cost. This is achieved through carefully smoothing the original nonsmooth regularizer (such as the ell_1 norm) hence reducing computational
iterations, and without violating the sample complexity bound for recovery. The paper is clearly written and the idea seems interesting, however, there are several practical issues that perhaps
should have been addressed.
The reviewer found the idea to squeeze more smoothing from excess observations quite interesting, however, this is at the expense of potentially compromising the recovery probability (eta in Fact
2.3, in particular the success recovery bound). A bit disappointingly, in the experiments the authors did not simulate the probability of "exact" recovery at all; instead, only some predefined
"closedness" to the true signal is used to declare victory of the algorithm. The authors did decrease the smoothing parameter by a factor of 4 to hedge against failure, but this is adhoc and
insufficient. It would be very interesting (in some sense also necessary) to see whether or not, or to what extent, does the aggressive smoothing affect the (exact) recovery probability. Judging from
the bound (Fact 2.3), there should be another tradeoff.
The experiments can be improved. The reviewer expected to see the following comparison: When we compare against some constant smoothing parameter, say mu = 0.1, we can compute the number of samples
needed for high probability recovery, call it m_mu, which is smaller than the number of available samples m. Then we just randomly throw away m - m_mu samples. This way one also reduces the
computational cost by reducing the size of the problem. In the reviewer's opinion, this should be the "conventional" constant smoothing algorithm to compare to. Right now, the authors seem to let the
competitor run on all available samples, putting it in a disadvantageous position. The two strategies (throwing away redundant data or aggressively smoothing) are really two blades of the same sword:
both aim at maintaining the recovery bound in the minimal sense. But the first approach is even more appealing: it certainly requires less memory. The reviewer strongly recommends the authors to
perform a serious comparison and report the relative strengths and weaknesses.
Another issue the reviewer would like to see addressed is practicality: so far the analysis and experiments are done with prior knowledge of the true signal. In practice this is not available. How
could one still be able to apply the aggressive smoothing without being too aggressive? On the other hand, it all appears to "tune" the smoothing parameter, which is what is done in practice anyways.
From a fully practical aspect, what is new and different here then? If the new message (as argued in line 325) is that the smoothing parameter needs to depend on the sample size, then how can this be
implemented without too much prior knowledge of the true signal?
Q2: Please summarize your review in 1-2 sentences
The paper did a good job in explaining and designing a strategy on how excess samples in regularized linear inverse problems can be exploited computationally to speed up convergence. The paper is
very well-written and the idea is interesting. However, there are several critical issues both theoretically and experimentally that perhaps should have been addressed.
Q1:Author rebuttal: Please respond to any concerns raised in the reviews. There are no constraints on how you want to argue your case, except for the fact that your text should be limited to a
maximum of 6000 characters. Note however, that reviewers and area chairs are busy and may not read long vague rebuttals. It is in your own interest to be concise and to the point.
We argue that we can reduce the computational cost of statistical optimization problems by aggressively smoothing when we have a large amount of data. While this idea seems very natural in
retrospect, we believe that it is a new and important principle. The case study presented provides theoretical and experimental evidence that such a time–data tradeoff exists in one instance. We
believe there are many other examples to explore, and we have identified several beyond the scope of the current work. In order to address the concerns raised during the review process, we submit our
answers to the three questions posed by the meta-reviewer.
Question 1: One reviewer mentions two papers that he or she believes reach very similar conclusions to our own. We will address these works and their relationship to our work specifically. We
believe, however, that our conclusion is new and different in character from the previous works. In particular, neither paper identifies a time–data tradeoff, nor can you derive one directly from
their results.
Lai and Yin indeed consider the sparse vector and low-rank matrix problems that we do in our case study. They propose choosing the smoothing parameter based on the l∞ norm of the signal (in the
sparse vector case) and the largest singular value of the signal (in the low-rank matrix case). We, however, calculate the location of the phase transition for exact recovery in the dual-smoothed
problem and choose a smoothing parameter based on the sample size as well. Our method results in a greater amount of smoothing and better performance as sample size increases. This is the time–data
Agarwal, Negahban and Wainwright show that some nonsmooth optimization problems in statistics have global linear convergence rates due to the properties of restricted strong convexity (RSC) and
restricted smoothness (RSM). They also show that the RSC/RSM parameters improve with sample size. They do not, however, discuss a time–data tradeoff explicitly. And while they show that iteration
count decreases as sample size increases, the overall computational cost rises (as evidenced by the constant smoothing scheme in Figures 3 and 4 of our work). That is, the properties of RSC/RSM alone
do not lead to a time–data tradeoff; the aggressive smoothing we impose does.
Question 2: One reviewer proposed that, given a fixed smoothing parameter, we should throw away excess samples in order to speed up the computation. In our framework, this is equivalent to simply
reducing the sample size from the outset. For the sparse vector case, the smallest sample size we test is m = 300. Since we cannot reduce m greatly in this case while guaranteeing exact recovery, we
feel justified in using m = 300 as a baseline value. Our result in Figure 3(b) clearly shows that our aggressive smoothing scheme, at every sample size m tested, has cost lower than the constant
smoothing scheme at the baseline m = 300. The same applies for the low-rank matrix experiment (Figure 4(b)). That is, throwing away excess samples under the constant smoothing scheme still results in
worse performance than our aggressive smoothing scheme.
The reviewer is indeed correct that the best course of action under the constant smoothing scheme would be to throw away data. Our principle, however, is to exploit excess samples to reduce
computational time via aggressive smoothing. The results of our experiments highlight the benefit of doing so (and the cost of not doing so).
Question 3: The case study we present relies on the Gaussian measurement assumption when calculating the location of the phase transition. There is evidence in the literature that problems of this
type exhibit some universality, and so it is reasonable to expect that other measurement matrices would work in practice.
One reviewer questions whether or not it is practical to compute the smoothing parameter without too much prior knowledge of the signal. A calculation of this parameter requires knowledge of the
sparsity (or rank) and magnitude of the signal, but an estimate will suffice. A conservative estimate will decrease the effectiveness of our method (by resulting in a less aggressive choice of the
smoothing parameter μ(m) as a function of the sample size m), but it will still adapt the smoothing to the sample size and perform better than smoothing using a small, constant parameter.
Furthermore, exact recovery will still hold.
Our principle of aggressive smoothing is practical, and it is not equivalent to adapting the smoothing parameter in an ad hoc manner to any given data set.
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Perspective (from Latin perspicere , ' see through, look through' ) describes the spatial, in particular linear, relationships between objects in space: the distance between objects in space in
relation to the location of the viewer. The perspective is always tied to the location of the viewer and can only be changed by changing the location of the objects and the viewer in the room.
This statement is important to the effect that a different perspective cannot be brought about by simply changing the viewing section without changing location (e.g. by using a zoom lens in
photography ).
The perspective representation summarizes the possibilities of depicting three-dimensional objects on a two-dimensional surface in such a way that a spatial impression is nevertheless created.
Types of perspective representation
• Geometric projection methods .
□ Central projection : Rays of vision emanate from one eye point, edges parallel to space seem to escape into a point in the projection - the so-called vanishing point .
☆ Central perspective (frontal perspective): Central projection with a vanishing point, one spatial area lies parallel to the image plane , this is depicted parallel to the
image , the other orthogonal to it - its spatial areas are aligned in one point
□ 2-point perspective (corner perspective): Central projection with two vanishing points, the horizontally parallel edges of the room are not parallel to the image plane and escape in their
respective vanishing point, the verticals are depicted parallel to the image.
□ Frog perspective : central projection with three vanishing points, there are no edges of the space parallel to the image plane, the eye point lies under the depicted object
□ Bird's eye view : Central projection with three vanishing points, there are no edges of the space parallel to the image plane, the eye point lies above the depicted object.
• Fisheye projection : spherical projection. Lines that do not go through the center are curved, areas at the edge are shown smaller than in the center of the picture, the viewing angle reaches 180
degrees and more.
• Parallel projection: Rays of vision run parallel, edges parallel to the room are also shown parallel in the projection. There are no vanishing points.
□ Orthogonal projection : Rays of sight hit the projection surface at right angles.
□ At an oblique angle: Rays of vision hit the projection surface at an oblique angle .
• Panorama image: With the panorama image, the image is first shown on a cylindrical surface, which can then be rolled up into a plane. But there are also large panoramas that are set up as
cylinders. Parallel lines are only shown in parallel in special cases. A viewing angle of 180 degrees and more (up to 360 degrees) can be achieved.
Perspective of meaning : Concept in painting. The size of the figures and objects depicted depends on their meaning, not on the spatial and geometric conditions.
Aerial and color perspective : the color and brightness contrasts decrease in the distance - colors appear more matt, mostly lighter and shifted into blue
Twisted perspective: first in the European cave paintings, later particularly noticeable in ancient Egyptian art , even later often in modern times, for example with Pablo Picasso .
The perspective representation of spatial situations can already be found in the first steps in the Franco-Cantabrian cave paintings , for example in the Grotto of Chauvet 30,000 years ago.
Perspective foreshortenings can also be found in other Upper Paleolithic caves, such as Trois Frères. Technically elaborate perspective processes were known to the Greeks, then to the Romans (see
also skenography ). Wall frescoes were found in Pompeii that were supposed to continue the space into a painted garden. In the centuries that followed, this knowledge was not further developed; early
Christian and medieval painting almost exclusively made use of the perspective of meaning , i.e. H. the size of the people and objects depicted was determined by their meaning in the picture, not by
their spatial arrangement. The spatial effect was achieved almost exclusively through the effect of the scenery , which differentiated a foreground level from a background.
During the Renaissance , the central perspective (in connection with the camera obscura ) was (re) discovered, which roughly corresponds to seeing with one eye or a distortion-free photographic
image. Painter architects such as Filippo Brunelleschi (considered the "inventor" of perspective) and Leon Battista Alberti created works that showed motifs from Christian iconography in spatially
correctly constructed architectural backdrops. In the 16th century, the artists and scholars of the Renaissance acquired extensive mathematical knowledge of perspectives and projections, which also
had an impact on the work of cartographers and the creation of cityscapes (especially initially in Italy). An early example of a geometrically exact and extremely detailed work of this kind is the
city view of Venice created by Jacopo de 'Barbari around 1500 .
Initially, the laws of the central perspective produced by our eyes were not recognized, and the representation was carried out by means of a cord that, starting from a fixed point, was stretched
over a simple grid in the form of a wire grid to the objects to be depicted . The draftsman sat next to the grid and transferred the measurements into the grid of his drawing surface ("perspective
constriction"). In a book from 1436 Leon Battista Alberti explained the mathematical methods with which a perspective effect can be achieved on paintings.
Albrecht Dürer published his book Underweysung der messung mit dem zirckel un richtscheyt in 1525 , which represented the first summary of the mathematical-geometric processes of central perspective
and thus also forms the basis of perspective construction processes as a sub-area of descriptive geometry .
Examples of perspective representations
Parallel perspective representation
Lines that run parallel in reality are also shown parallel in parallel perspective imaging. This prevents the lines from converging in the direction of the vanishing points, so that the areas shown
remain clearly visible. This effect is z. B. desired by architects who aim to ensure that the views of houses are always the same regardless of the perspective. Architects speak of “parallel
perspective” here.
Axonometric representation
Axonometric representations are parallel perspective representations. The vanishing point has moved to infinity. The axonometric projections include isometric and dimetric representations.
Isometric axonometry, according to DIN 5
If the body to be displayed is rotated 45 ° in the top view and lifted at the rear in the side view so that its surface is less than approx. 35.26 ° (exactly ) to the base, a three-dimensional image
is projected with the height (H) perpendicular , the lengths (L) and depths (T) appear at an angle of 30 ° to the baseline. The directions of width (length), height and depth expansion (in the case
of a right-angled body, the directions of the edges) appear at different angles, but not at right angles to each other. All three directions are equally shortened, are related to each other like 1:
1: 1, so they have a common scale (isometry). Starting from a vertical spatial coordinate (axis, hence "axonometry"), all edges or points of a body are only constructed using the spatial coordinates
that are actually at right angles to one another, perpendicular or at an angle of 30 ° to the base line. Lines or edges that do not have the direction of a spatial coordinate (e.g. the sloping gable
edge of a house, the diagonals of a cube) are not shown to scale, so they cannot be constructed directly. The end points of such lines have to be determined via a detour using the spatial
coordinates. It is useful to draw three line scales in the direction of the spatial coordinates in such a representation in order to make it clear that the dimensions are correct only in these
directions. This spatial representation is preferable for bodies with equivalent views. It is called isometric axonometry according to DIN 5 (see fig.). ${\ displaystyle \ arctan (1 / {\ sqrt {2}})}$
Dimetric axonometry, according to DIN 5
If the body to be displayed is rotated by only 20 ° in the top and side view, a three-dimensional image is created in which the lengths are displayed at an angle of 7 ° and the depths at an angle of
42 ° to the base line. The depths appear shortened by half compared to the heights and lengths. The spatial image is therefore two-scale (dimetric, e.g. H and L 1: 5, T 1:10). This representation is
used in a front view that is particularly important compared to the other views. Taking into account the various angles and scales, the body is otherwise drawn as in isometric axonometry. To make
this clear, it is essential to draw in the appropriate line scales in the form of a cross of the spatial axes. According to DIN 5, this method of representation is called Dimetric Axonometry (see
Keystone projection
The oblique projection is also a parallel projection . In contrast to axonometric methods, two axes can be left undistorted here, and only the third axis is shown obliquely and (possibly) shortened.
Examples are the cavalier perspective and the cabinet perspective . With the former, the elevation is undistorted and lines that run perpendicular to it are shown unabridged, with the latter, these
lines are shortened by half (as shown in the picture above).
Another name for a special type of oblique projection is the military perspective. As in the cavalier perspective, there is no shortening of the third axis. The floor plan is applied undistorted and
vertical routes are shown true to scale.
Central perspective representation
The simplest form of perspective is the central perspective . It is mainly used in architecture and for illustration. Edges parallel to space are not shown parallel to the image, but optically
combine in an apparent, imaginary point, the so-called vanishing point . This vanishing point lying on the horizon line can be found via the interface that is created by extending the object edges
that are parallel in reality.
The central perspective is a special form of the vanishing point perspective in which the vanishing point is in the center of the image. This usually gives a front view of the object. There are no
shifts to the right or left, but up or down. Even if several vanishing points arise due to different object edges, such as when depicting a house, they all lie on the horizon line. The surfaces of
the object facing the viewer are parallel to the image, while the object edges leading into the depths of the space seem to unite in a vanishing point on the horizon .
Further variants are the perspectives with two - also called corner perspectives - or three vanishing points. Since with a perspective with three vanishing points the horizon necessarily moves
upwards or downwards, the respective images are also called frog's eye or bird's eye view .
Cylindrical projection
Various artists such as B. MC Escher have experimented with other variants of the perspective, such. B. the cylindrical projection. With this perspective, panoramas of 180 ° and more can be
realistically represented, but straight lines distort into curved curves. An example of this is Escher's lithograph Staircase I from 1951 (with “Krempeltierchen”).
Relief perspective representation
This type of perspective does not lead to a complete 2-D representation, but only greatly shortens one dimension of the 3-D space. In doing so, the appearance of the objects viewed from a fixed point
of view does not change, since objects lying behind are also correspondingly reduced in size with an exact relief .
Meaning perspective representation
In the time before the rediscovery of the geometric perspective, the so-called meaning perspective was used in panel paintings. The size and orientation of the people shown in the picture depends on
their importance: important protagonists appear large, less important ones appear smaller, even if they are spatially in front of the other person. In the picture example on the right, the
quasi-isometric perspective of the footstools only relates to the respective figure - from a graphical-compositional point of view, this arrangement enables the (flat) opening of the picture space to
the background. The perspective of meaning is already used in ancient Egyptian art: while the pharaoh and wife are shown in full size, slaves and courtiers are shown much smaller. In the icon
painting , this type of presentation can be found as well as in the painting of the Romanesque and Gothic . The perspective of meaning can still be found today in naive painting .
Aerial and color perspective representation
A distinction must be made between aerial and color perspective .
In the aerial perspective, an impression of depth is created in that the contrasts decrease from front to back and the brightness increases from front to back. Regardless of the color, a sharp /
unsharp contrast is created at the same time as the contours that become more indistinct towards the rear.
The color perspective creates an impression of depth by using different colors in the foreground, middle and background. Warm colors dominate in the foreground (yellow, orange, red, brown), while
colder green and blue tones dominate the middle ground and background. Instead, there may also be a green or blue cast.
Multi-perspective representation
A multi-perspective describes a spatial representation using several projection centers or the combination of different perspectives that result in a space. In addition to the use of
multi-perspective in still images, it is also used in digital space on moving images. By using several projection centers in moving images, distortions can be avoided. (Example: A sphere retains its
circular shape, and an elliptical shape of the sphere due to the camera distortion is avoided.) The multi-perspective (in the still image) can be found in various forms of art history in various
paintings. A very pronounced application of multi-perspective can be found in the works of the Swiss painter and media artist Matthias AK Zimmermann . For example, his painting “The Frozen City”
shows a panoramic landscape, the space and objects of which arise from various perspectives ( central perspective , cavalier perspective , military perspective , suggestion of a fish-eye lens ,
Experience perspective
In the experiential perspective, the artists endeavor to reproduce what they see through close observation. This brings you very close to the central perspective and also recognizes that objects in
the background are blurring and becoming more bluish ( color perspective ). Experience perspective stands for an approximately correct vanishing point representation before Italian artists succeeded
just a few years later in constructing the central perspective geometrically perfect.
See also
Web links
Individual evidence
1. ↑ ^a ^b plane geometric projections ( page no longer available , search in web archives ) Info: The link was automatically marked as defective. Please check the link according to the instructions
and then remove this notice. on a website of the University of Tübingen, with further sources.
2. ↑ Müller-Karpe, p. 195; Leroi-Gourhan, p. 132 ff.
3. ↑ Chauvet, p. 114.
4. ↑ Müller-Karpe, p. 197.
5. ^ Leon Battista Alberti: Della Pittura - About the art of painting . Ed .: Oskar Bätschmann and Sandra Gianfreda. Knowledge Buchgesellschaft, Darmstadt 2002, ISBN 3-534-15151-8 .
6. ↑ Albrecht Dürer: Underweysung the measurement with the Zirckel and Richtscheyt. Verlag A. Wofsy, Nuremberg, June 1981. ISBN 0-915346-52-4 .
7. ↑ Geometric illustration practice - from painting to computer graphics ( Memento of the original from January 28, 2015 in the Internet Archive ) Info: The archive link has been inserted
automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 5 MB).
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How to Fit Camera to Object
Asked by Cayden Serrano on
6 Answers
Answer by Kamdyn Moore
A dynamic width/height of the canvas if the screen resizes,Several predefined camera Vector3 positions,A scene containing several Object3D instances,A user can select a camera position (from above)
If the camera is centered and viewing the cube head-on, define
dist = distance from the camera to the _closest face_ of the cube
height = height of the cube.
If you set the camera field-of-view as follows
fov = 2 * Math.atan( height / ( 2 * dist ) ) * ( 180 / Math.PI ); // in degrees
EDIT: If you want the cube width to match the visible width, let aspect be the aspect ratio of the canvas ( canvas width divided by canvas height ), and set the camera field-of-view like so
fov = 2 * Math.atan( ( width / aspect ) / ( 2 * dist ) ) * ( 180 / Math.PI ); // in degrees
Source: https://stackoverflow.com/questions/14614252/how-to-fit-camera-to-object
Answer by Preston Robbins
View -> Align View -> Align Active Camera to Selected.,Select the objects you wish to put in the camera view.,Or run the operator: bpy.ops.view3d.camera_to_view_selected(),I am not at all a
programmer, but have you looked into the View All operator?
import bpy
from bpy import context
# Select objects that will be rendered
for obj in scene.objects:
obj.select = False
for obj in context.visible_objects:
if not (obj.hide or obj.hide_render):
obj.select = True
Source: https://blender.stackexchange.com/questions/51563/how-to-automatically-fit-the-camera-to-objects-in-the-view
Answer by Cohen Parra
Hi there! I want to create a function zoomToFit(camera, targetObject), which makes the camera move and zoom to fit the target object.,Summary: I want to zoom perspective camera to fit target object
width, without changing fov.,I do not want to modify the fov of the camera, instead I want move the camera, in order to create the zoom,To do this I will move the camera in the direction of
camera.lookAt(Object) vector; however, I cannot figure out the rest of the math… and I have researched a lot, without success…
Note that there are a couple of expectations - the object should be “in front” of the camera, in particular.
const fitCameraToObject = function ( camera, object, offset, controls ) {
offset = offset || 1.25;
const boundingBox = new THREE.Box3();
// get bounding box of object - this will be used to setup controls and camera
boundingBox.setFromObject( object );
const center = boundingBox.getCenter();
const size = boundingBox.getSize();
// get the max side of the bounding box (fits to width OR height as needed )
const maxDim = Math.max( size.x, size.y, size.z );
const fov = camera.fov * ( Math.PI / 180 );
let cameraZ = Math.abs( maxDim / 4 * Math.tan( fov * 2 ) );
cameraZ *= offset; // zoom out a little so that objects don't fill the screen
camera.position.z = cameraZ;
const minZ = boundingBox.min.z;
const cameraToFarEdge = ( minZ < 0 ) ? -minZ + cameraZ : cameraZ - minZ;
camera.far = cameraToFarEdge * 3;
if ( controls ) {
// set camera to rotate around center of loaded object
controls.target = center;
// prevent camera from zooming out far enough to create far plane cutoff
controls.maxDistance = cameraToFarEdge * 2;
} else {
camera.lookAt( center )
Source: https://discourse.threejs.org/t/camera-zoom-to-fit-object/936
Answer by Rafael Park
You can set the camera's position, field-of-view, or both.,I am assuming you are using a perspective camera.,At this point, you can back the camera up a bit, or increase the field-of-view a bit.,If
you set the camera field-of-view as follows
If the camera is centered and viewing the cube head-on, define
dist = distance from the camera to the _closest face_ of the cube
height = height of the cube.
If you set the camera field-of-view as follows
fov = 2 * Math.atan( height / ( 2 * dist ) ) * ( 180 / Math.PI ); // in degrees
EDIT: If you want the cube width to match the visible width, let aspect be the aspect ratio of the canvas ( canvas width divided by canvas height ), and set the camera field-of-view like so
fov = 2 * Math.atan( ( width / aspect ) / ( 2 * dist ) ) * ( 180 / Math.PI ); // in degrees
Based on WestLangleys answer here is how you calculate the distance with a fixed camera field-of-view:
dist = height / 2 / Math.tan(Math.PI * fov / 360);
To calculate how far away to place your camera to fit an object to the screen, you can use this formula (in Javascript):
// Convert camera fov degrees to radians
var fov = camera.fov * ( Math.PI / 180 );
// Calculate the camera distance
var distance = Math.abs( objectSize / Math.sin( fov / 2 ) );
Here's a CodePen showing this in action. The relevant lines:
var fov = cameraFov * ( Math.PI / 180 );
var objectSize = 0.6 + ( 0.5 * Math.sin( Date.now() * 0.001 ) );
var cameraPosition = new THREE.Vector3(
sphereMesh.position.y + Math.abs( objectSize / Math.sin( fov / 2 ) ),
Source: https://newbedev.com/how-to-fit-camera-to-object
Answer by Dalton Barajas
Given an object being viewed and the camera position they have chosen how do I compute the final camera position to "best fit" the object on screen?,If the camera positions are used "as is" on some
screens the objects bleed over the edge of my viewport whilst others they appear smaller. I believe it is possible to fit the object to the camera frustum but haven't been able to find anything
suitable.,The following calculation is exact for an object that is a cube, so think in terms of the object's bounding box, aligned to face the camera.,If the camera is centered and viewing the cube
head-on, define
If the camera is centered and viewing the cube head-on, define
dist = distance from the camera to the _closest face_ of the cube
height = height of the cube.
If you set the camera field-of-view as follows
fov = 2 * Math.atan( height / ( 2 * dist ) ) * ( 180 / Math.PI ); // in degrees
EDIT: If you want the cube width to match the visible width, let aspect be the aspect ratio of the canvas ( canvas width divided by canvas height ), and set the camera field-of-view like so
fov = 2 * Math.atan( ( width / aspect ) / ( 2 * dist ) ) * ( 180 / Math.PI ); // in degrees
Source: https://coderedirect.com/questions/56688/how-to-fit-camera-to-object
Answer by Jolie Alexander
We currently provide Box3.setFromObject( object ) from which the user can get the AABB. The user can then set the camera position/fov/zoom as desired.,compute new camera parameters (eg lookat,
position),Some users might not like that solution, however, as it involves moving the camera to a new place.,Some users might not like that solution, however, as it involves moving the camera to a
new place.
dist = height / 2*tg( (fov/2) * (Pi/180) )
Source: https://github.com/mrdoob/three.js/issues/6784
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Matching pennies in class
Yesterday I took the first class of my new module on Game Theory. I've been very excited about this as I think Game Theory is a really fun subject to learn (and teach).
In class we covered the first two chapters of my notes (
Chapter 1: Introduction to Game Theory
Chapter 2: Normal Form Games
). Whilst talking about Normal Form Games I showed the students the game called Matching Pennies.
Two players each show a coin with either 'Heads' or 'Tails' showing. If both coins match then the 1st player (the row player) wins, otherwise the 2nd player (the column player) wins.
This can be represented using a 'bi-matrix':
Each tuple of that matrix corresponds to a pair of strategies from the set $\{H, T\}$, so if the row player chose $H$ and the column player chose $T$ then they would read the outcome in the
first row
second column:
$(-1,1)$. The convention used here is that outcomes show the
to the first and then the second player. So in this instance the row/first player would get -1 and the column/second player would get 1 (ie the column player wins because the coins where different).
I asked the students to get in to pairs and record five rounds of the game on some paper (
forms and all other content for the course available at this github repo
After that, I modified the game to give this:
The row player still wins when the coins match but there is just more to win/lose when $H$ is picked by the row player.
I got the students to once again record the results.
Last night I got home and instead of speaking to my wife I went through and entered all the data.
Here are some of the results.
First of all 'basic matching pennies'. Here are the moving averages of all the games played:
I'm graphing the probability with which players played $H$. As you can see 'we' got to equilibrium pretty quickly and 'on average' players were randomly swapping between $H$ and $T$.
Here is a plot of the equivalent mean score to both players:
First of all we see that the plots are reflections in the $x=0$ line of each other. This is because the game we are considering is called a Zero Sum Game: all the utility doublets sum to 0. Secondly
we see that the mean score is coming around to 0.
All of the above is great and more or less exactly what you would expect.
While playing the second game I overheard a couple of students say something like 'Oh this is a bit more complicated: we need to think'.
They were completely right!
Here are the results. First of all the strategies:
It seems like students are once again playing with equal probabilities of picking $H$ or $T$. The outcome for the score is again very similar:
Is this what we expect?
Not quite.
Let us assume row players are playing a 'mixed strategy' $\sigma_1=(x,1-x)$ (ie they choose $H$ with probability $x$) and column players are playing $\sigma_2=(y,1-y)$.
Let us see what the expected utility to the
player is when $\sigma_2=(.5,.5)$ as a function of $x$ (the probability of playing $H$):
So in fact what the row player does is irrelevant (with regards to his/her utility) as long as the column player plays $\sigma_2=(.5,.5)$.
What about the column player?
Writing down the utility to the column player when $\sigma_1=(.5,.5)$ as a function of $y$ (the probability of playing $H$):
So NOW if $\sigma_1=(.5,.5)$ it looks like the column player has SOME control over his/her utility.
Here is a plot of that $u_2$:
So our plot is that of a decreasing function. Remember $y$ is something that the column player can control. So as the column player wants to increase $u_2$: the best response they should adapt to the
row player playing $\sigma_1=(.5,.5)$ is in fact $y^*=0$ because at $y=0$ the utility is at it's highest!
What this implies (for the second game) is that whilst the students were all winning and losing in equal measure (the mean score was around 0). The column player could in fact improve their strategy
and take advantage of the fact that the row player was playing $\sigma_1=(.5,.5)$.
The row player can't actually do this (we did the math above and we saw that he/she couldn't really have any effect on his/her utility). What my students and I will see in
Chapter 6
of my class is that in fact there is a way to make both players 'unable to improve their outcomes'. When we get there it will also shed light on the dashed lines in some of the plots of this blog
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Nonlinear optical response of three-level systems
We investigate the properties of a homogeneously broadened three-level system interacting with two optical fields, E[ω1] (x,t) and E[ω2] (x,t), with central frequencies ω[1] and ω[2] whose amplitudes
are assumed to be slowly varying in time compared with dephasing times of the optical transitions. We calculate Raman susceptibilities allowing each transition to be induced by both fields and
without employing the rotating wave approximation (RWA); this enables correct calculation of the optical response of the medium when the fields are on-resonance or arbitrarily far off-resonance with
the Raman transition. In some frequency regions the RWA causes substantial errors. A distinction is drawn between two different types of RWA: (a) a neglect of off-resonant components of the
polarization for direct optical transitions, and (b) neglect of off-resonant components of off-diagonal density matrix elements of order E [ω1] (x,t) E[ω2] (x,t)* in the field amplitudes. The first
RWA is known to become inaccurate when the pump frequency is significantly off-resonance with the pump transition. We show that the second RWA becomes inaccurate when |ω[1] - ω[2]| is small or
comparable to the active Raman mode frequency (even when the field are close to resonance). Calculations based upon our expressions show that when the pump frequency is close to resonance with the
optical transition from the ground state, a nonlinear absorption at the Stokes shifted frequency occurs. This absorption can significantly decrease the Raman gain. Also, we find that Rabi splittings
in a three-level system can occur even when the pulse durations of the incident fields are short compared with the T[1] times of the system. We solve analytically for the optical response of the
system to two fields of arbitrary strength and frequency. We analyze this solution and its perturbative expansion in the field amplitudes. We use the Maxwell-Bloch equations developed here to
propagate time-dependent pulses.
ASJC Scopus subject areas
• General Physics and Astronomy
• Physical and Theoretical Chemistry
Dive into the research topics of 'Nonlinear optical response of three-level systems'. Together they form a unique fingerprint.
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This page has worksheets and activity mats with 5-frames, 10-frames, and double 10-frames. Get Started. Number bonds like times tables are something which a child should know instantly without
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of the worksheets displayed are Math background, 1st grade addition with base 10 blocks, Partnership elements work, Lessonlesson make 10 to add, Icd 10 putting codes into practice, Exercise the
valued directions work, W o r k s h e e t s, Partners adjusted basis … English tenses: worksheets, printable exercises pdf, handouts to print. Kindergarten and 1st Grade. b) Who has the least
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What are Solid Shapes ⭐ Definition, Types, Properties, Examples
Solid Shapes – Definition With Examples
Updated on January 3, 2024
In the captivating world of mathematics and geometry, one concept that stands out due to its wide-ranging application and intriguing complexity is the curved line. A curved line, unlike a straight
line, bends and twirls, changing its direction at every point on its path. From the graceful arcs of rainbows kids enjoy drawing to the elliptical orbits of planets, the concept of curved lines
permeates our daily lives and the cosmos alike.
At Brighterly, we believe in nurturing curiosity and fostering a love for learning in children. It’s our mission to take complex topics like curved lines and present them in an engaging,
understandable way that sparks intrigue in young minds. This article is part of our effort to make mathematics an enjoyable journey for children, taking them on a tour of the fascinating world of
curved lines, where they can witness the harmonious dance between the abstract world of numbers and the physical world around us.
Solid Shapes: Introduction
Let’s embark on an exciting journey to the world of solid shapes. What’s this realm, you ask? Well, it’s around us every day! The chair you’re sitting on, the ball you kick around, the ice cream cone
you enjoy on a summer day, all these are examples of solid shapes. They’re a central part of our lives, playing a crucial role not only in our daily activities but also in advanced mathematics and
geometry. Unraveling the secret world of these shapes will not only enhance your knowledge but also provide you with a new perspective to understand and appreciate the world around you. You’ll also
be ahead in your mathematics class with Brighterly.
What are Solid Shapes in Geometry?
Solid shapes in geometry are three-dimensional figures that have length, breadth, and height. Unlike two-dimensional shapes like squares or circles, which are flat, solid shapes extend in three
directions. They possess depth, giving them a form we can hold, touch, and explore in reality. They make up objects we use, admire, and interact with daily, like a football, a skyscraper, or a tiny
Understanding solid shapes can be thrilling, akin to being an explorer discovering new lands. When we study these shapes, we look at their attributes, such as their faces, edges, and vertices. Every
different solid shape has its unique properties, much like every country has its own unique culture and landscape. Let’s dive deeper into the intriguing world of solid shapes!
Elements of Solid Shapes
The intriguing attributes of solid shapes, namely faces, edges, and vertices, are like their DNA – unique and distinctive. A face is a flat or curved surface on a solid shape. An edge is a line
segment where two faces meet, and a vertex is a point where three or more edges meet. Understanding these elements is like learning a new language, a language that helps us communicate, understand,
and design the spatial world around us. And, when you comprehend this language well, you can effectively engage with the incredible field of geometry!
Solid Shapes and Their Properties
Now that we have a basic understanding of solid shapes and their elements, we must investigate their individual properties. Every solid shape has a unique set of attributes – this includes their
number of faces, edges, vertices, the calculation of their surface area, and their volume. They are like different species in a vast jungle, each carrying their own fascinating traits.
Types of Solid Shapes
Our world is filled with an array of solid shapes, each having their own identity. Let’s explore some common types, such as spheres, cylinders, cuboids, cubes, cones, pyramids, and prisms. These are
not merely names but are like keys that unlock various secrets of mathematics. Each one of these shapes has its characteristics, formulas, and unique properties that we will delve into.
A sphere is a perfect example of a solid shape. It’s round, smooth, and doesn’t have edges or vertices. A real-life example? Imagine a perfectly round ball or the Earth (if we overlook its minor
Properties of a Sphere
Being the smoothest of solid shapes, a sphere is uniquely characterized by its center and radius. Unlike other shapes, it has no edges, no vertices, and only one face, which is curvilinear. All
points on the surface of a sphere are equidistant from the center, and this distance is known as the radius of the sphere.
Surface Area of a Sphere
The surface area of a sphere is the total area that its surface covers. It’s calculated using the formula 4πr², where r is the radius of the sphere. For example, if the radius of a sphere is 5 units,
the surface area will be 4π(5)² or 100π square units.
Volume of a Sphere
The volume of a sphere is the amount of space it occupies, and it is given by the formula (4/3)πr³. So, for a sphere with a radius of 5 units, the volume would be (4/3)π(5)³ or 500/3π cubic units.
Think of a can of your favorite drink, and you have a perfect example of a cylinder. A cylinder is a solid shape with two parallel circular faces (the bases) and one curved face that connects the
Properties of a Cylinder
A cylinder has 3 faces, 2 edges, and no vertices. The parallel circular faces are identical in size, and the distance between them is called the height of the cylinder.
Surface Area of a Cylinder
The surface area of a cylinder can be found using the formula 2πrh + 2πr², where r is the radius of the base and h is the height of the cylinder.
Volume of a Cylinder
The volume of a cylinder is calculated as πr²h. So if we know the radius and height of a cylinder, we can easily find how much space it occupies.
A cuboid is what most people think of when they hear the term ‘box’. It has six faces, all of which are rectangles, and it has 12 edges and 8 vertices.
Properties of a Cuboid
A cuboid is characterized by its length, breadth, and height. All faces are at right angles to each other, and the opposite faces of a cuboid are equal.
Surface Area of a Cuboid
The surface area of a cuboid can be found using the formula 2(lb + bh + hl), where l is the length, b is the breadth, and h is the height of the cuboid.
Volume of a Cuboid
The volume of a cuboid is calculated by multiplying its length, breadth, and height (lbh). It represents the amount of space that the cuboid occupies.
A cube is a unique shape in the world of solid shapes. Imagine a perfectly shaped dice, and you have a cube. It’s a special type of cuboid where all faces are square, and all edges are of equal
Properties of a Cube
A cube has 6 faces, 12 edges, and 8 vertices. All faces of a cube are squares of equal size, and all its edges are of the same length. Moreover, all angles in a cube are right angles, and each face
meets its four neighboring faces at equal angles of 90 degrees.
Surface Area of a Cube
The surface area of a cube can be determined using the formula 6a², where a is the length of the edge. If the edge of the cube is 4 units, for instance, its surface area will be 6(4)² or 96 square
Volume of a Cube
The volume of a cube, i.e., the amount of space it occupies, is given by the formula a³. So, if the edge of a cube measures 4 units, its volume would be 4³ or 64 cubic units.
When you think of a cone, think of an ice cream cone. It’s a solid shape with a circular base and a curved surface that tapers to a point, called the apex or the vertex of the cone.
Properties of a Cone
A cone has 1 face, 1 edge, and 1 vertex. The face is a circle (the base of the cone), and the edge is a curved line, forming the curved surface that connects the base with the vertex.
Surface Area of a Cone
The surface area of a cone is found using the formula πr(r + l), where r is the radius of the base, and l is the slant height of the cone.
Volume of a Cone
The volume of a cone represents the space it occupies and can be calculated by the formula (1/3)πr²h, where r is the radius of the base, and h is the height of the cone.
A pyramid is a solid shape that has a polygonal base and triangular faces that meet at a common vertex. Picture the famous Egyptian pyramids, and you’ll get an idea of this shape.
Properties of a Pyramid
The properties of a pyramid vary depending on the shape of the base. A pyramid always has one face more than the number of sides on the base polygon. It also has as many vertices and edges as the
base polygon has sides.
Surface Area of a Pyramid
The surface area of a pyramid can be found by adding the area of the base to the sum of the areas of each triangular face. The formula differs depending on the base shape.
Volume of a Pyramid
The volume of a pyramid is given by the formula (1/3)Bh, where B is the area of the base, and h is the height of the pyramid.
A prism is a fascinating solid shape with two identical polygonal bases and rectangular faces that connect corresponding vertices of the bases. Picture a box of cereal, which is an example of a
rectangular prism.
Properties of a Prism
The properties of a prism depend on the nature of the bases. However, all prisms have an equal number of faces, vertices, and edges as the polygon of the bases. For instance, a triangular prism has 5
faces, 9 edges, and 6 vertices.
Surface Area of a Prism
The surface area of a prism is calculated by adding the areas of its bases to the areas of its rectangular faces. It is generally given by the formula 2B + Ph, where B is the area of a base, P is the
perimeter of a base, and h is the height of the prism.
Volume of a Prism
The volume of a prism, which represents the amount of space it occupies, is determined by the formula Bh, where B is the area of a base and h is the height of the prism.
Faces, Edges, and Vertices of Solid Shapes
Understanding the faces, edges, and vertices of solid shapes is like holding a decoder ring for 3D geometry. These key elements provide a foundation for identifying, classifying, and comparing
various solid shapes. Let’s delve deeper into what each of these elements is.
Faces of Solid Shapes
In the context of solid shapes, a face is a flat or curved surface. For instance, a cube has 6 square faces, while a sphere has a single curved face.
Edges of Solid Shapes
An edge is a line segment where two faces of a solid shape meet. A cuboid, for example, has 12 edges, while a sphere has no edges.
Vertices of Solid Shapes
A vertex is a point where three or more edges meet. A cone has one vertex at the tip, while a cylinder has no vertices.
Practice Problems on Solid Shapes
1. What is the volume of a cube with an edge of 6 units?
2. Calculate the surface area of a cylinder with a radius of 4 units and a height of 5 units.
3. Find the volume of a cone with a base radius of 3 units and a height of 7 units.
4. If a rectangular prism has a length of 4 units, a width of 3 units, and a height of 2 units, what is its surface area?
As we reach the conclusion of this exploration into the world of solid shapes, we hope your child’s understanding of 3D shapes and their properties has expanded. The beauty of mathematics lies not
just in the realm of numbers but also in the visual, tangible world of geometry.
Here at Brighterly, we believe that by exploring these mathematical concepts in a fun, engaging, and accessible way, we can inspire a lifelong love for learning. And we understand that every child is
unique, which is why we strive to create resources that are tailored to meet different learning styles.
Remember, mastering solid shapes is not a one-day affair. So, keep revisiting these concepts, practice with the problems provided, and before you know it, your child will be a whiz in geometry!
Frequently Asked Questions on Solid Shapes
What is the difference between 2D shapes and solid shapes?
2D shapes, or two-dimensional shapes, have length and width, but no thickness. They are flat and can only be measured in two directions, such as a square, a circle, or a triangle. On the other hand,
solid shapes are three-dimensional (3D). They have length, width, and height, giving them volume and allowing them to occupy space, like a cube, sphere, or a cylinder.
Why are vertices important in solid shapes?
Vertices are where the edges of a shape meet. They are significant because they give us vital information about the structure of a shape. Counting the vertices, along with faces and edges, helps us
identify, classify, and describe the solid shape.
What is the relationship between the faces, edges, and vertices in a cube?
A cube has 6 faces, 12 edges, and 8 vertices. This corresponds with Euler’s formula for polyhedra, which states that for any convex polyhedron (including a cube), the number of vertices (V) plus the
number of faces (F) is equal to the number of edges (E) plus 2. So for a cube, V + F = E + 2 becomes 8 + 6 = 12 + 2, which indeed holds true.
How is the volume of a sphere calculated?
The volume of a sphere is given by the formula (4/3)πr³, where r is the radius of the sphere. This formula essentially tells us how much space the sphere occupies.
What real-world objects are examples of prisms?
Prisms are everywhere in our world! A book, a box of cereal, a tent, or a Toblerone chocolate bar can be seen as examples of prisms. These everyday objects can help children understand and relate to
the concept of prisms in a more practical and enjoyable way.
Information Sources:
Poor Level
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Nathan's Notes
Mamba: Selective State Space Modeling
An introduction to Mamba models: faster and better* than transformers
Jul 22, 2024
This is a blog post regarding a talk I gave at K-Scale Labs to help everyone understand Mamba. The slides are here. State Space Models (SSMs) are one of the few machine learning archetypes that is
competitive with Mamba in both inference speed and effectiveness — not to mention, they have some pretty cool intuition!
Papers + Resources
Some sources to look for more insight + math:
What’s the problem with transformers?
There are a few:
• During qkv, calculating the attention score of each token (relative to every other) is $O(N^2)$ according to the length of the context window.
• Further, the quadratically-growing key-value cache needs to be stored alongside the model during inference.
• Moreover, having a context window at all is rather limiting
If we can address these problems with a different architecture, we can hopefully be better at long-range tasks, synthesizing speech and video (which are packed with context), training models quicker,
and inference on smaller devices.
What are these “State Space Models (SSMs)”?
SSMs are characterized by two equations \begin{aligned} x'(t) &= Ax(t) + Bu(t) \\ y(t) &= Cx(t) + Du(t) \end{aligned}
However, we can simply interpret this as a function-to-function mapping $u(t) \mapsto y(t)$ parameterized by $A$, $B$, $C$, and $D$ (fixed “latent” parameters). Then, $x(t)$ is a latent
representation satisfying the ODE. The important aspect of SSMs is that they have three views:
1. Continuous: This is simply our original equation — a function-to-function mapping of $u(t) \mapsto y(t)$ following the equation denoted above.
2. Recurrent: We can map our linear ODE to discrete steps in an RNN, with similar intuition for how we approach Flow Matching (see slide 4). We essentially turn our function-to-function mapping into
a sequence-to-sequence mapping through interpolation
\begin{aligned} x_k &= Ax_{k-1} + Bu_k \\ y_k &= Cx_k + Du_k \end{aligned}
1. Convolutional: We unroll our recurrent view, solving $N$ steps ahead with the recurrence to get a kernel of size $N$, replugging-in values such that
$x_0 = \overline{B}u_0 \quad x_1 = \overline{A}B u_0 + \overline{B}u_1 \quad x_2 = \overline{A}^2 B u_0 + \overline{A}B u_1 + \overline{B}u_2 \quad \cdots$
Each of these views have tradeoffs, but the key version we want to focus on is the recurrent view. It has effectively no context window (unlike the convolutional view), which addresses the issue we
have with transformers. On the other hand, it has a efficient constant time inference (unlike the continuous view), which once again addresses another problem we had with transformers.
However, recurrent networks are not easily parallelizable in training, as each inference is dependent on the previous time-step. Additionally, we’re at risk of vanishing/exploding gradients. How do
we fix these problems?
Let’s drop the state space model idea — at this point, SSMs are an RNN that has its roots in continuous space and the possibility to become a convolution. Contrarily, an RNN models information
statefully by being able to merge all of its previously seen context into one state.
Linear RNNs
Having an activation function makes our calculations way too complicated and difficult to parallelize. What we really want is a way to calculate the accumulation of all an RNN’s layer multiplications
quickly and easily. For each iteration with
$h_t = f_W(h_{t-1}, x_t)$
we can’t possibly expect condense these calculations quickly with a nonlinear activation function like $\tanh$ Instead, let’s remove the activation function so we’re left with
$f_W(h, x) = W_h h + W_x x$
How do we parallelize this? If we think about the Blelloch algorithm for parallel prefix sum, we can find some inspiration. With Blelloch, we are able to take the prefix sum of an array of length $n$
in $O(\log(N))$ sequential steps through parallelization simply because the addition property is associative. We do this by accumulating the sum over different step sizes.
Luckily there’s an associative function we can utilize for our new activiationless RNN layers.
$f((W_1, x_1), (W_2, x_2)) = (W_1 W_2, W_1 x_1 + x_2)$
Now, we are able to apply this function do our recurrence in $O(N\log(N))$ (we are folding in $W_h$ and $W_x$ together). But, there is an additional problem where we have to store $W_i W_j \dots W_k$
during the scanning which results in a very large cubic cache as we have to store a ${\rm I\!R}^{d \times d}$ matrix for each input during the Blelloch scan. Luckily, because $W_i \in {\rm I\!R}^{d \
times d}$, we can easily diagonalize these intermediate products and keep our cache quadratic.^1
Boom! Now we have a quick and parallelizable RNN. To add nonlinearity (which we still want), we can simply add an additional layer to our RNN which is applied to the recurrent part’s outputs (just
like the feed forward layer after self-attention layers).
Avoiding Exploding/Vanishing Gradients
RNNs are incredibly sensitive and any small errors could result in exploding/vanishing gradients such that the our model never converges. Further, we can’t make use of gradient truncation as it would
superficially make our model short range.
The solution is how we initialize our weights. If we make them very small, our gradients will be small as well. The creators of Linear RNN initialize each parameter $w$ such that
$w = e^{-e^a}e^{ib}, e^{-e^a} \sim \text{Uniform}([0.999, 1.0]), b \sim \text{Uniform}([0, \frac{\pi}{10}])$
On the other hand, our parameters are also very sensitive to inputs. We want to reduce their magnitude as much as possible. So, we multiply each input by $\Delta = \sqrt{1 - e^{-e^a}}$. From there,
we’re done! ^2
Mamba improves upon using one key idea: selection. With just a recurrent network, after many iterations, it is very easy to have to hold too much information in $h_t$. The RNN should have some idea
of attention — what should be retained in $h_t$ and what should be removed?
The eloquent solution, as expected, is gates. We want $W_h$ and $W_x$ (preserving notation from traditional RNNs) to be dependent on $x_i$ itself. Below shows the exact psuedocode for an SSM
The most important thing about this algorithm is that the “selecting” functions and latent parameters^4 (specifically $A$ and $B$) can be chosen such that $A=-1$, $B=1$, $s_\Delta = \text{Linear}
(x_t)$, and $\tau_\Delta = \text{softplus}$ where the result is that each channel of our RNN having a gate is characterizable by
\begin{aligned} g_t &= \sigma(\text{Linear}(x_t)) \\ h_t &= (1 - g_t)h_{t-1} + g_tx_t \end{aligned}
This provides us an intuitive gate and shows us that we are really just considering how much we want to “remember” the current input at each timestep. Now, we can selectively recall information in
very long context windows. Not to mention, our recurrence is only $O(N \log N)$ with the parallelization step.^5 Now, we have Mamba!
1. Something pretty cool — $P$ and $P^{-1}$ in the diagonalization $PDP^{-1}$ are actually learned matrices to add more expressivity, while not having to worry about any matrix-inverting problems. ↩
2. SSMs, in aims to convert the continuous-time view into a recurrent view, are remotely the same but with a different initialization scheme. In such, weights are initialized where $w = e^{\Delta
(a+bi)}$ and inputs are multiplied by $(\Delta(a+bi))^{-1}(w-1) \circ \Delta$ where $\Delta \in [0.1, 0.0001]$. Otherwise, our intuition that the recurrent SSM view is the same as Linear RNN
holds. ↩
3. Notice that the parameter $D$ is ignored because it is easily computable as a skip connection. ↩
4. The Mamba paper offers interesting intuition regarding each of the parameters in the 3.5.2 Interpretation of Selection Mechanisms section. ↩
5. It’s pretty interesting to explore handware-aware understanding of runtime. There are cool optimizations such as recomputing intermediate steps during the backward pass for backpropagation in
order to reduce cache size during training. Additionally, similar to FastAttention, we want to do all our heavy linear algebra (recurrence) on SRAM, where this data transfer ends up being the
largest bottleneck to training speed more than the recurrence itself. ↩
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Papers with Code - Ronen Basri
no code implementations • 30 Jan 2017 • Onur Ozyesil, Vladislav Voroninski, Ronen Basri, Amit Singer
The structure from motion (SfM) problem in computer vision is the problem of recovering the three-dimensional ($3$D) structure of a stationary scene from a set of projective measurements, represented
as a collection of two-dimensional ($2$D) images, via estimation of motion of the cameras corresponding to these images.
Motion Estimation Simultaneous Localization and Mapping +1
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Core module: Automata theory and formal languages
This module is taught in English. This module is regularly offered every summer term.
This module was previously offered in the summer terms 2022 and 2023.
Automata and formal languages are classic topics in theoretical computer science, related to mathematical logic. Automata are abstract machines to solve computational problems, by automatically
executing a predetermined sequence of steps. Automata can thus also be seen as finite representations of formal languages, which model computational problems and may be infinite. Formal languages are
classified according to the Chomsky hierarchy, which provides the overarching theme of this module. In turn, formal languages are generated by grammars, which are the third major object (next to
automata and formal languages) which are studied in this module. We will study the theories which link these three objects, and study their important roles in fundamental applications such as
compiler construction, formal verification, and parsing in natural and computer languages.
Recommended prerequisites
We expect attendees to be proficient in the material from the following modules:
• Discrete Algebraic Structures
• Mathematics I
• Mathematics II
We further recommend basic knowledge of the following topics:
• Programming in Python, C++, C# or Java
Learning goals
In this module you will learn
• to design automata for specific tasks,
• to design formal languages and classify them according to the Chomsky hierarchy,
• to link specific automata, formal languages and grammars in terms of their expressive power,
• to model computational problems and decide the (non-)existence of finite algorithms for their solution.
In this module we will cover the following topics:
• Basics of automata and formal languages
• The Chomsky hierarchy
• Deterministic finite automata, definition and construction
• Regular languages, closure properties, word problem, string matching
• Nondeterministic automata: Rabin-Scott transformation of nondeterministic into deterministic automata
• Epsilon automata, minimization of automata, elimination of e-edges, uniqueness of the minimal automaton (modulo renaming of states)
• Myhill-Nerode Theorem: Correctness of the minimization procedure, equivalence classes of strings induced by automata
• Pumping Lemma for regular languages: provision of a tool which, in some cases, can be used to show that a finite automaton principally cannot be expressive enough to solve a word problem for some
given language
• Regular expressions vs. finite automata: Equivalence of formalisms, systematic transformation of representations, reductions
• Pushdown automata and context-free grammars: Definition of pushdown automata, definition of context-free grammars, derivations, parse trees, ambiguities, pumping lemma for context-free grammars,
transformation of formalisms (from pushdown automata to context-free grammars and back)
• Chomsky normal form
• CYK algorithm for deciding the word problem for context-free grammrs
• Deterministic pushdown automata
• Deterministic vs. nondeterministic pushdown automata: Application for parsing, LL(k) or LR(k) grammars and parsers vs. deterministic pushdown automata, compiler compiler
• Regular grammars: Turing machines and linear bounded automata vs general and context-sensitive grammars
• Turing machines
• Halting problem
• Weekly lectures (in presence) and weekly tutorial session (in presence).
• Each week an assignment is handed out, which should be solved individually outside the classroom and then be submitted for grading. The assignment and its solutions are then discussed in the
subsequent tutorial session. If sufficiently many points are awarded in the grading process (cumulative over all assignments of the term), then a bonus is given for the final exam at the end of
the term.
• Final exam (written) for 90 minutes at the end of the term, whose outcome (together with a potential bonus from the assignment) determines the final grade for the module.
Overall module load: 6 ETCS credit points.
Provided materials
- Lecture slides
- Lecture recordings
- Assignment sheets
- Simulator software
Recommended literature
• John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation. Pearson, 3rd edition, 2006
• Uwe Schöning: Logik für Informatiker. Spektrum, 5th edition, 2000
• Martin Kreuzer, Stefan Kühling: Logik für Informatiker. Pearson Studium, 2006
• Gottfried Vossen, Kurt-Ulrich Witt: Grundkurs Theoretische Informatik, Vieweg-Verlag, 2010.
• Christel Baier, Joost-Pieter Katoen: Principles of Model Checking, The MIT Press, 2007
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Module Belt.Map
The top level provides generic immutable map operations.
It also has three specialized inner modules Belt.Map.Int, Belt.Map.String and
Belt.Map.Dict: This module separates data from function which is more verbose but slightly more efficient
A immutable sorted map module which allows customize compare behavior.
The implementation uses balanced binary trees, and therefore searching and insertion take time logarithmic in the size of the map.
For more info on this module's usage of identity, `make` and others, please see the top level documentation of Belt, A special encoding for collection safety.
Example usage:
module PairComparator = Belt.Id.MakeComparable(struct
type t = int * int
let cmp (a0, a1) (b0, b1) =
match Pervasives.compare a0 b0 with
| 0 -> Pervasives.compare a1 b1
| c -> c
let myMap = Belt.Map.make ~id:(module PairComparator)
let myMap2 = Belt.Map.set myMap (1, 2) "myValue"
The API documentation below will assume a predeclared comparator module for integers, IntCmp
Specalized when key type is int, more efficient than the generic type, its compare behavior is fixed using the built-in comparison
specalized when key type is string, more efficient than the generic type, its compare behavior is fixed using the built-in comparison
This module seprate identity from data, it is a bit more verboe but slightly more efficient due to the fact that there is no need to pack identity and data back after each operation
('key, 'identity) t
'key is the field type
'value is the element type
'identity the identity of the collection
The identity needed for making an empty map
let make: id:id('k, 'id) => t('k, 'v, 'id);
make ~id creates a new map by taking in the comparator
let m = Belt.Map.make ~id:(module IntCmp)
let isEmpty: t(_, _, _) => bool;
isEmpty m checks whether a map m is empty
isEmpty (fromArray [|1,"1"|] ~id:(module IntCmp)) = false
let has: t('k, 'v, 'id) => 'k => bool;
has m k checks whether m has the key k
has (fromArray [|1,"1"|] ~id:(module IntCmp)) 1 = true
let cmpU: t('k, 'v, 'id) => t('k, 'v, 'id) => Js.Fn.arity2(('v => 'v => int)) => int;
let cmp: t('k, 'v, 'id) => t('k, 'v, 'id) => ('v => 'v => int) => int;
cmp m0 m1 vcmp
Total ordering of map given total ordering of value function.
It will compare size first and each element following the order one by one.
let eqU: t('k, 'v, 'id) => t('k, 'v, 'id) => Js.Fn.arity2(('v => 'v => bool)) => bool;
let eq: t('k, 'v, 'id) => t('k, 'v, 'id) => ('v => 'v => bool) => bool;
eq m1 m2 veq tests whether the maps m1 and m2 are equal, that is, contain equal keys and associate them with equal data. veq is the equality predicate used to compare the data associated with the
let findFirstByU: t('k, 'v, 'id) => Js.Fn.arity2(('k => 'v => bool)) => option(('k, 'v));
let findFirstBy: t('k, 'v, 'id) => ('k => 'v => bool) => option(('k, 'v));
findFirstBy m p uses funcion f to find the first key value pair to match predicate p.
let s0 = fromArray ~id:(module IntCmp) [|4,"4";1,"1";2,"2,"3""|];;
findFirstBy s0 (fun k v -> k = 4 ) = option (4, "4");;
let forEachU: t('k, 'v, 'id) => Js.Fn.arity2(('k => 'v => unit)) => unit;
let forEach: t('k, 'v, 'id) => ('k => 'v => unit) => unit;
forEach m f applies f to all bindings in map m. f receives the 'k as first argument, and the associated value as second argument. The bindings are passed to f in increasing order with respect to the
ordering over the type of the keys.
let s0 = fromArray ~id:(module IntCmp) [|4,"4";1,"1";2,"2,"3""|];;
let acc = ref [] ;;
forEach s0 (fun k v -> acc := (k,v) :: !acc);;
!acc = [4,"4"; 3,"3"; 2,"2"; 1,"1"]
let reduceU: t('k, 'v, 'id) => 'acc => Js.Fn.arity3(('acc => 'k => 'v => 'acc)) => 'acc;
let reduce: t('k, 'v, 'id) => 'acc => ('acc => 'k => 'v => 'acc) => 'acc;
reduce m a f computes (f kN dN ... (f k1 d1 a)...), where k1 ... kN are the keys of all bindings in m (in increasing order), and d1 ... dN are the associated data.
let s0 = fromArray ~id:(module IntCmp) [|4,"4";1,"1";2,"2,"3""|];;
reduce s0 [] (fun acc k v -> (k,v) acc ) = [4,"4";3,"3";2,"2";1,"1"];;
let everyU: t('k, 'v, 'id) => Js.Fn.arity2(('k => 'v => bool)) => bool;
let every: t('k, 'v, 'id) => ('k => 'v => bool) => bool;
every m p checks if all the bindings of the map satisfy the predicate p. Order unspecified
let someU: t('k, 'v, 'id) => Js.Fn.arity2(('k => 'v => bool)) => bool;
let some: t('k, 'v, 'id) => ('k => 'v => bool) => bool;
some m p checks if at least one binding of the map satisfy the predicate p. Order unspecified
let size: t('k, 'v, 'id) => int;
size s
size (fromArray [2,"2"; 2,"1"; 3,"3"] ~id:(module IntCmp)) = 2 ;;
let toArray: t('k, 'v, 'id) => array(('k, 'v));
toArray s
toArray (fromArray [2,"2"; 1,"1"; 3,"3"] ~id:(module IntCmp)) = [1,"1";2,"2";3,"3"]
let toList: t('k, 'v, 'id) => list(('k, 'v));
let fromArray: array(('k, 'v)) => id:id('k, 'id) => t('k, 'v, 'id);
fromArray kvs ~id
toArray (fromArray [2,"2"; 1,"1"; 3,"3"] ~id:(module IntCmp)) = [1,"1";2,"2";3,"3"]
let keysToArray: t('k, 'v, 'id) => array('k);
keysToArray s
keysToArray (fromArray [2,"2"; 1,"1"; 3,"3"] ~id:(module IntCmp)) =
let valuesToArray: t('k, 'v, 'id) => array('v);
valuesToArray s
valuesToArray (fromArray [2,"2"; 1,"1"; 3,"3"] ~id:(module IntCmp)) =
let minKey: t('k, _, _) => option('k);
let maxKey: t('k, _, _) => option('k);
let minimum: t('k, 'v, _) => option(('k, 'v));
let maximum: t('k, 'v, _) => option(('k, 'v));
let get: t('k, 'v, 'id) => 'k => option('v);
get s k
get (fromArray [2,"2"; 1,"1"; 3,"3"] ~id:(module IntCmp)) 2 =
Some "2";;
get (fromArray [2,"2"; 1,"1"; 3,"3"] ~id:(module IntCmp)) 2 =
let getWithDefault: t('k, 'v, 'id) => 'k => 'v => 'v;
getWithDefault s k default
See get
let getExn: t('k, 'v, 'id) => 'k => 'v;
getExn s k
See getExn
raise when k not exist
let remove: t('k, 'v, 'id) => 'k => t('k, 'v, 'id);
remove m x when x is not in m, m is returned reference unchanged.
let s0 = (fromArray [2,"2"; 1,"1"; 3,"3"] ~id:(module IntCmp));;
let s1 = remove s0 1;;
let s2 = remove s1 1;;
s1 == s2 ;;
keysToArray s1 = [|2;3|];;
let removeMany: t('k, 'v, 'id) => array('k) => t('k, 'v, 'id);
removeMany s xs
Removing each of xs to s, note unlike remove, the reference of return value might be changed even if none in xs exists s
let set: t('k, 'v, 'id) => 'k => 'v => t('k, 'v, 'id);
set m x y returns a map containing the same bindings as m, with a new binding of x to y. If x was already bound in m, its previous binding disappears.
let s0 = (fromArray [2,"2"; 1,"1"; 3,"3"] ~id:(module IntCmp));;
let s1 = set s0 2 "3";;
valuesToArray s1 = ["1";"3";"3"];;
let updateU: t('k, 'v, 'id) => 'k => Js.Fn.arity1((option('v) => option('v))) => t('k, 'v, 'id);
let update: t('k, 'v, 'id) => 'k => (option('v) => option('v)) => t('k, 'v, 'id);
update m x f returns a map containing the same bindings as m, except for the binding of x. Depending on the value of y where y is f (get x m), the binding of x is added, removed or updated. If y is
None, the binding is removed if it exists; otherwise, if y is Some z then x is associated to z in the resulting map.
let mergeMany: t('k, 'v, 'id) => array(('k, 'v)) => t('k, 'v, 'id);
mergeMany s xs
Add each of xs to s, note unlike set, the reference of return value might be changed even if all values in xs exist s
let mergeU: t('k, 'v, 'id) => t('k, 'v2, 'id) => Js.Fn.arity3(('k => option('v) => option('v2) => option('v3))) => t('k, 'v3, 'id);
let merge: t('k, 'v, 'id) => t('k, 'v2, 'id) => ('k => option('v) => option('v2) => option('v3)) => t('k, 'v3, 'id);
merge m1 m2 f computes a map whose keys is a subset of keys of m1 and of m2. The presence of each such binding, and the corresponding value, is determined with the function f.
let keepU: t('k, 'v, 'id) => Js.Fn.arity2(('k => 'v => bool)) => t('k, 'v, 'id);
let keep: t('k, 'v, 'id) => ('k => 'v => bool) => t('k, 'v, 'id);
keep m p returns the map with all the bindings in m that satisfy predicate p.
let partitionU: t('k, 'v, 'id) => Js.Fn.arity2(('k => 'v => bool)) => (t('k, 'v, 'id), t('k, 'v, 'id));
let partition: t('k, 'v, 'id) => ('k => 'v => bool) => (t('k, 'v, 'id), t('k, 'v, 'id));
partition m p returns a pair of maps (m1, m2), where m1 contains all the bindings of s that satisfy the predicate p, and m2 is the map with all the bindings of s that do not satisfy p.
let split: t('k, 'v, 'id) => 'k => ((t('k, 'v, 'id), t('k, 'v, 'id)), option('v));
split x m returns a tuple (l r), data, where l is the map with all the bindings of m whose 'k is strictly less than x; r is the map with all the bindings of m whose 'k is strictly greater than x;
data is None if m contains no binding for x, or Some v if m binds v to x.
let mapU: t('k, 'v, 'id) => Js.Fn.arity1(('v => 'v2)) => t('k, 'v2, 'id);
let map: t('k, 'v, 'id) => ('v => 'v2) => t('k, 'v2, 'id);
map m f returns a map with same domain as m, where the associated value a of all bindings of m has been replaced by the result of the application of f to a. The bindings are passed to f in increasing
order with respect to the ordering over the type of the keys.
let mapWithKeyU: t('k, 'v, 'id) => Js.Fn.arity2(('k => 'v => 'v2)) => t('k, 'v2, 'id);
let mapWithKey: t('k, 'v, 'id) => ('k => 'v => 'v2) => t('k, 'v2, 'id);
mapWithKey m f
The same as map except that f is supplied with one more argument: the key
let getData: t('k, 'v, 'id) => Belt__.Belt_MapDict.t('k, 'v, 'id);
getData s0
Advanced usage only
let getId: t('k, 'v, 'id) => id('k, 'id);
getId s0
Advanced usage only
let packIdData: id:id('k, 'id) => data:Belt__.Belt_MapDict.t('k, 'v, 'id) => t('k, 'v, 'id);
packIdData ~id ~data
Advanced usage only
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Syllabus Information
Use this page to maintain syllabus information, learning objectives, required materials, and technical requirements for the course.
Syllabus Information
MTH 106 - Math in Society 2
Associated Term: Spring 2023
Learning Objectives: Upon successful completion of this course, the student will be able to:
1. Apply learning
a. Use mathematics and quantitative reasoning to solve problems
b. Show appropriate mathematical mechanics and techniques
c. Apply skills and abilities to new situations in and out of the classroom
d. Demonstrate skills and proficiencies in algebraic manipulation and calculator use
2. Think critically
a. Choose an appropriate solution strategy
b. Construct a mathematical plan to solve a problem
c. Determine the reasonableness and implications of mathematical solutions
d. Recognize the limitations of mathematical models
3. Communicate effectively a. Clarify and explain thought process and solution
b. Explain results orally and/or in writing
c. Collaborate with others to solve problems effectively
d. Interpret results and solutions
e. Justify reasoning and solution
4. Engage diverse values with civic and ethical awareness
a. Evaluate how various decisions would fit with personal values and civic awareness
b. Practice decision-making in authentic settings with realistic numbers
5. Create ideas and solutions
a. Reflect on successes, failures, and obstacles encountered in the problem-solving process
b. Assess mistakes and rework solutions
Required Materials:
Technical Requirements:
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What is 60% of 25? [Solved] | Brighterly Questions
What is 60% of 25?
Updated on January 9, 2024
Answer: 15
Percentage Calculations
Calculating 60% of 25 involves converting the percentage to a decimal (0.60) and then multiplying it by 25. The result is 0.60×25=15. Understanding how to calculate percentages of numbers is a
fundamental skill in mathematics and is widely used in real-life scenarios such as determining discounts, calculating tips, and understanding statistical data. Being able to quickly calculate
percentages of numbers is a valuable skill in managing finances and making informed decisions during shopping and budgeting. Mastery of percentage calculations is essential in finance, retail, and
data analysis, where such computations are frequent and crucial.
FAQ on Practical Percentages
What is 50% of 30?
What is 25% of 40?
What is 75% of 20?
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What is the difference between displacement and distance calculus?
What is the difference between displacement and distance calculus?
The displacement of a particle moving in a straight line is a vector defined as the change in its position. If the particle moves from the position x(t1) to the position x(t2), its displacement is x
(t2)−x(t1) for the time interval [t1,t2]. The distance travelled by a particle is the ‘actual distance’ travelled.
What is displacement in integrals?
To find the displacement (position shift) from the velocity function, we just integrate the function. The negative areas below the x-axis subtract from the total displacement. Displacement = To find
the distance traveled we have to use absolute value.
What is the relationship between distance and displacement?
Distance is the length of the path covered to reach from initial to final positions. And displacement is the shortest or linear distance between the initial and the final positions. So, distance is
greater than or equal to displacement.
Is displacement the integral of velocity?
The definite integral of a velocity function gives us the displacement. To find the actual distance traveled, we need to use the speed function, which is the absolute value of the velocity.
What are the 5 difference between distance and displacement?
However, displacement takes both the magnitude and direction of the path travelled by an object. Hence, distance is a scalar quantity and displacement is a vector quantity. Distance is always
positive or zero, while displacement can be positive, negative or zero.
What is displacement in calculus?
The displacement is simply the difference in the position of the two marks and is independent of the path taken when traveling between the two marks. The distance traveled, however, is the total
length of the path taken between the two marks.
When can displacement be equal to distance?
The magnitude of displacement is equal to the distance when the motion is in a fixed direction (one direction ).
How do you find the integral of displacement and velocity?
Velocity is the derivative of displacement. Integrate velocity to get displacement as a function of time. We’ve done this before too. The resulting displacement-time relationship will be our second
equation of motion for constant jerk ….constant jerk.
a = a0 + jt [1]
s = s0 + v0t + ½a0t2 + ⅙jt3 [2]
a = f(s) [4]
What is difference between distance and displacement explain with diagram?
Explain with the help of a diagram….
DISTANCE DISPLACEMENT
1) It is the actual path covered by an object. It is the shortest distance covered by an object.
2) It is a scalar quantity. It is a vector quantity.
3) It cannot be negative. It can be negative.
What is the difference between displacement and difference?
The basic difference between Distance and displacement is that distance is the length of a path between two points and displacement is the shortest distance between two points….What is Distance?
Distance Displacement
It is considered a scalar magnitude. It is considered a vector magnitude.
How are displacement and distance similar and different?
Distance is a scalar quantity that refers to “how much ground an object has covered” during its motion. Displacement is a vector quantity that refers to “how far out of place an object is”; it is the
object’s overall change in position.
Can displacement be greater than distance?
Displacement can’t be greater than Distance.
Under what conditions distance and displacement are equal?
Can displacement be greater than the distance?
Is displacement the derivative of velocity?
The first derivative of displacement is velocity. The second derivative of displacement is acceleration.
What are derivatives of displacement?
Absement is the integral of displacement;
Absity is the double integral of displacement;
Abseleration is the triple integral of displacement;
Abserk is the fourth integral of displacement;
Absounce is the fifth integral of displacement,and so on
What is the formula of displacement?
The word displacement implies that an object has moved, or has been displaced. Displacement is defined to be the change in position of an object. It can be defined mathematically with the following
equation: Displacement = Δ x = x f − x 0 text {Displacement}=Delta x=x_f-x_0 Displacement=Δx=xf−x0.
How can I calculate the magnitude of displacement?
First,enter the value of the Initial Velocity and choose the unit of measurement from the drop-down menu.
Then enter the value of the Final Velocity and choose the unit of measurement from the drop-down menu.
Finally,enter the value of the Time and choose the unit of measurement from the drop-down menu.
How to find displacement calculus?
Displacement calculate is find three way. (1):- When you know only final position value and initial position value Displacement (Δx) = xf – xi. (2):- When you know inital velocity value, acceleration
of object and time then used this formula Displacement (Δx) = ut + 1 / 2 at² Note:- this formula also used when you know velocity and time 👍 Δx = ut ,acceleration is zero.
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Mean Curvature - (Calculus IV) - Vocab, Definition, Explanations | Fiveable
Mean Curvature
from class:
Calculus IV
Mean curvature is a measure of the curvature of a surface at a point, defined as the average of the principal curvatures at that point. It provides insight into how a surface bends in space and is
essential in understanding the behavior of surfaces in differential geometry, particularly when relating to arc length and curvature.
congrats on reading the definition of Mean Curvature. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Mean curvature is calculated as the average of the two principal curvatures at a point on the surface.
2. A surface with zero mean curvature is known as a minimal surface, which means it locally minimizes area.
3. In 3-dimensional space, mean curvature can be positive, negative, or zero, indicating whether the surface curves outward, inward, or is flat at that point.
4. Mean curvature plays a crucial role in physical applications, such as soap films and biological membranes, where surfaces tend to minimize their area.
5. The concept of mean curvature extends beyond surfaces to higher dimensions, where it helps define geometric properties of manifolds.
Review Questions
• How is mean curvature defined and what significance does it hold for understanding surfaces?
□ Mean curvature is defined as the average of the principal curvatures at a given point on a surface. It is significant because it helps describe how a surface bends in space, offering insights
into its geometric properties. By analyzing mean curvature, we can identify minimal surfaces and understand physical phenomena like how soap films behave by minimizing their surface area.
• Compare and contrast mean curvature with Gaussian curvature. How do both contribute to our understanding of surfaces?
□ Mean curvature provides an average measure of bending at a point on a surface, while Gaussian curvature represents a product of the principal curvatures, capturing intrinsic geometric
properties. While mean curvature can indicate local behavior (like whether a surface is minimal), Gaussian curvature relates more to the overall shape of the surface. Together, they give a
comprehensive view of how surfaces interact with space and provide critical information for applications in differential geometry.
• Evaluate the role of mean curvature in physical phenomena like soap films and biological membranes. How does this understanding impact real-world applications?
□ Mean curvature plays a vital role in understanding physical phenomena such as soap films and biological membranes because these structures tend to minimize their area due to surface tension.
By analyzing mean curvature, scientists can predict how these surfaces will behave under various conditions, leading to advancements in materials science and biology. This understanding has
practical applications in designing efficient membrane technologies and studying cellular structures, demonstrating how geometric principles impact real-world scenarios.
"Mean Curvature" also found in:
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Calculus - early transcendentals, metric edition - Startsidan
[📖PDF] Calculus: Early Transcendentals 0538497904 Ladda
Köp boken Single variable calculus - early transcendentals (9781305272385) hos Borås Studentbokhandel - Finns i butiken, skickas inom 24h 8:e upplagan, 2015. Köp Calculus, Early Transcendentals,
International Metric Edition (9781305272378) av James Stewart på campusbokhandeln.se. 1133710883 | Essential Calculus: Early Transcendentals | This book is for instructors who think that most
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Stewart's "Calculus: Early Transcendentals, Fifth Edition" has the mathematical precision, accuracy, clarity of exposition and outstanding examples and problem sets that have characterized the first
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Choose from 28 different sets of Calculus Early Transcendentals Stewart View Solutions for Calculus: Early Transcendentals. stewartet07 image. Title: Publisher: Author: Calculus: Early
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File Type: PDF. Language: English. Calculus: early transcendentals. Collapse menu 1 Analytic Geometry. 1. Lines; 2 The book includes some exercises and examples from Elementary Calculus:
Calculus: Early Transcendentals £180.84 Only 1 left in stock. Stewart's "Calculus: Early Transcendentals, Fifth Edition" has the mathematical precision, accuracy, clarity of exposition and
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With CALCULUS: EARLY TRANSCENDENTALS, Seventh Edition, Stewart conveys not only the utility of calculus to help you develop technical competence, but also gives you an appreciation for the intrinsic
beauty of the subject. ABOUT THIS TEXTBOOK – Calculus: Early Transcendentals, originally by D. Guichard, has been redesigned by the Lyryx editorial team. Substantial portions of the content,
examples, and diagrams have been redeveloped, with additional contributions provided by experienced and practicing instructors. Calculus: Early Transcendentals by James Stewart. I II III IV V VI VII
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There you will find a review, with examples and exercises, of the very basic rules of algebra that are needed for success in calculus. When I grade calculus exams, I find that the mistakes my
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of Calculus: Early Transcendentals. The website has been designed to give you easy access to study materials, book supplements and challenge problems that will help you with your study of calculus.
Look in this section for information about new material, notices and resources posted In the “early transcendentals” method, the logarithmic and exponential functions are introduced shortly after the
definition of the derivative.
Substantial portions of the content, examples, and diagrams have been redeveloped, with additional contributions provided by experienced and practicing instructors. Calculus: Early Transcendentals
CDN$ 182.84 In Stock. Success in your calculus course starts here!
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NCERT Solutions for Class 6 Maths Updated for 2024-25
NCERT Solutions for Class 6 Maths Chapter-Wise PDF Free Download
NCERT Solutions for Class 6 Maths includes extensive explanations for all of the problems in the NCERT textbooks recommended by the Central Board of Secondary Education. Vedantu offers
chapter-specific NCERT Class 6 Maths Solutions to help students solve their questions by providing a thorough comprehension of the subjects. Our team of experts built this in a well-structured format
to give the finest techniques of problem solving and to ensure good conceptual knowledge. These resources, including NCERT Solutions, are accessible in PDF format, allowing students to download and
learn offline as well.
1. NCERT Solutions for Class 6 Maths Chapter-Wise PDF Free Download
2. Chapter-Specific NCERT Solutions for Class 6 Maths
3. Glance on NCERT Solutions Class 6 Maths | Vedantu
4. NCERT Solutions for Class 6 Maths Chapter Details, Formulas, and Exercises PDF
4.1Chapter 1 Knowing Our Numbers
4.2Chapter 2 Whole Numbers
4.3Chapter 3 Playing with Numbers
4.4Chapter 4 Basic Geometrical Ideas
4.5Chapter 5 Understanding Elementary Shapes
4.9Chapter 9 Data Handling
4.10Chapter 10 Mensuration
4.12Chapter 12 Ratio and Proportion
5. CBSE Class 6 Maths Chapter-Wise Marks Weightage
6. CBSE Class 6 Maths Study Materials
Chapter-Specific NCERT Solutions for Class 6 Maths
Given below are the chapter-wise NCERT Solutions for Class 6 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.
Note: The chapters on Symmetry and Practical Geometry have been excluded from the Class 6 Maths textbook for the 2024-25 academic year
Glance on NCERT Solutions Class 6 Maths | Vedantu
• NCERT Solutions for Class 6 Maths for all the chapters and exercises from Chapters 1 to 12 are provided.
• Practising the textbook questions using these solutions can help students analyse their level of preparation and understanding of concepts.
• Covering chapters like Knowing Our Numbers, Whole Numbers, Playing with Numbers, Basic Geometrical Ideas, Understanding Elementary Shapes, Integers, Fractions and more.
• This page provides details about the exam pattern, marks weightage, and question paper design for CBSE Class 6 Maths.
• This article also provides resources such as NCERT notes, important questions, exemplar solutions, RD Sharma, and RS Aggarwal solutions PDF for further reference.
FAQs on NCERT Solutions for Class 6 Maths 2024-25
1. Why Should you Opt for NCERT Solutions for Class 6 Maths?
A subject like Mathematics might not be a favourite of most students. However, with NCERT solutions for CBSE Class 6 Maths, students will be compelled to find the subject interesting and find
motivation in preparing for its exams. These study materials will not only help students in clearing doubts but also improve their problem-solving and time management capabilities.
2. How can I prepare for my board exams with NCERT Class 6 Maths Solution pdf?
All you need to do is visit the official website of Vedantu and you will have your sights on the PDF version of the solutions provided by our experts. You can practice these solutions, again and
again, to manage time better and have an idea about the marking scheme and question pattern.
3. Does NCERT Maths Book Class 6 Solutions pdf free download help students a lot?
Yes, absolutely. Students can have a lot of help from the solutions of Vedantu. All the answers provided in the PDF are very accurate and verified. Plus, Vedantu experts have formulated the solutions
according to the guidelines of CBSE. So, students will be able to gain a lot of insights into the exam pattern and answering format. Also, the solutions have been explained in an easy-to-understand
format. Thus, students can easily comprehend the concepts and practice regularly to understand the chapters better. Go for NCERT Maths Book Class 6 Solutions PDF Free Download today.
4. How to download NCERT Class 6 Maths Solution pdf?
Students will be able to download the NCERT Class 6 Maths Solution PDF from the official website of Vedantu.
Students can download the NCERT Class 7 Maths Solutions from Vedantu and that too chapter-wise. All the solutions have been provided in a PDF format on the basis of the chapters. All they have to do
is log in to Vedantu and choose the specific Class, Subject, and Chapter that they want to download. This way, they will easily be able to find the study materials and start their preparation for the
5. Why should I use the NCERT Solutions for Class 6 Maths?
Students should use the NCERT Solutions for Class 6 Maths provided by Vedantu as these are easy to understand. You can use the NCERT Maths Book Class 6 Solutions PDF free download to understand the
concepts easily. You can find answers to all exercises and learn the formulas properly. You can download the PDFs for Class 6 Maths NCERT Solutions and can refer to them anytime you want. These
solutions are free of cost and prepared by experts, so they are 100% reliable.
6. In Class 6 Maths, which are the most important topics to focus for exams?
There are a total of 14 chapters in Class 6 Maths. Some chapters need more practice. Students can refer to NCERT Solutions for CBSE Class 6 Maths for all chapters at Vedantu Website or the Vedantu
app. All solutions are prepared by subject matter experts and explained well for a better understanding of students. Chapter 10 Mensuration and Chapter 12 Ratio and Proportion are difficult chapters
and might need more practice. Therefore, students need to practice more questions of these two chapters to score good marks.
7. Is it sufficient if I study only from the Class 6 Maths NCERT book?
Yes, if students study and practice the questions given in the Class 6 Maths NCERT book, it is sufficient and they can score high marks in Class 6 exams. NCERT books are prepared by subject matter
experts and are accurate. They follow the latest CBSE guidelines and exam patterns. Students should practice all questions thoroughly. It is important to understand the concepts for scoring high
marks in the Maths exam. To check the solutions students can use the NCERT Maths Book Class 6 Solutions PDF free download.
8. Why should students solve Class 6 Maths NCERT Questions?
Questions given in the Maths NCERT book for Class 6 can help students to practice and understand the concepts. Students can solve Class 6 Maths NCERT Solutions and practice all questions from the
book. They should understand the concepts and solve questions two or three times to score good marks in their Maths exam. Vedantu provides Class 6 Maths NCERT Solutions for the convenience of
students. The solutions can help students to practice maths daily.
9. Are NCERT Solutions for Class 6 Maths provided by Vedantu free of cost?
Yes, NCERT Solutions for Class 6 Maths provided by Vedantu are free of cost. Students just need to download them on their computers. Visit the page on NCERT Maths Book Class 6 Solutions PDF free
download on the Vedantu website (vedantu.com). Click on the chapter you want the solutions for. Solutions are available in Hindi and English medium. Click on the download PDF to download the
solutions. You can save them on your computers and refer to them whenever you have doubts or queries.
10. Is class 6th Maths difficult?
Maths in class six will vary in complexity depending on a few aspects, so it won't be for everyone.
• New ideas: Compared to previous grades, sixth grade covers new ideas such fractions, decimals, and fundamental algebra. If these are unfamiliar to you, they may be difficult.
• Foundation Matters: Mastering addition, subtraction, multiplication, and division with whole numbers will make it easier for you to understand these new ideas.
• Learning Style: While some students have an easier time understanding new mathematical concepts, others do not.
11. How to score 100 in Maths class 6?
While achieving a perfect score in CBSE Class 6 Maths demands commitment and concentration, the following advice can help you do far better:
• Understand the Ideas: Don't only commit formulas to memory. Make an effort to comprehend the "why" underlying every idea. This will enable you to use them more successfully in many contexts and
improve your word problem-solving techniques.
• Learn the Fundamentals: Make sure you understand addition, subtraction, multiplication, and division of whole numbers, fractions, and decimals thoroughly. Proficiency in these will be essential
for tackling increasingly intricate issues.
• Master Arithmetic: Make sure you can perform addition, subtraction, multiplication, and division with whole numbers with ease. Everything else in Class 6 Maths is built upon this.
• Tables of Practice: Accuracy in your computations will increase and you'll save time if you can multiplication tables up to at least 12x12.
12. What are the general objectives of Maths class 6?
The Class 6 NCERT Maths overarching goals are to reinforce fundamental abilities and present fresh ideas in order to get pupils ready for more advanced maths. Here's a summary of some important
Sense of Numbers:Improve knowledge of whole numbers, including their properties (divisibility, even/odd numbers), as well as their operations (addition, subtraction, multiplication, and division).
Expand your knowledge to include conversions and operations with decimals and fractions (proper, improper, and mixed).
Algebraic Thinking: Gain a rudimentary knowledge of variables, which are unknowns, and how expressions represent them.
Present the idea of solving for the unknown variable and basic equations.
Geometry: Develop a basic understanding of the geometric shapes (circles, quadrilaterals, angles, lines, and triangles).
• Discover the characteristics of these forms, such as triangle angles and various triangle kinds.
• Learn how to create and recognise geometric shapes with the aid of instruments such as compasses and rulers.
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How do you make a polynomial fit in Excel?
You can use the LINEST() function in Excel to fit a polynomial curve with a certain degree. The function returns an array of coefficients that describes the polynomial fit.
How does Excel fit polynomial trendline?
Then you can simply add a trendline. Select polynomial in this example we use a fourth order polynomial display the equation press close and there we have our fourth order polynomial fitted to the
Can you solve polynomial in Excel?
Finding Root of Polynomial Using Microsoft Excel
Firstly, input the coefficient of the equations in separate cells (B4:E4). 2. Then, you may guess the value of x with any number as initial value (B6). 3.
How do you make a trendline polynomial in Excel?
Add a trendline
1. Select a chart.
2. Select the + to the top right of the chart.
3. Select Trendline. Note: Excel displays the Trendline option only if you select a chart that has more than one data series without selecting a data series.
4. In the Add Trendline dialog box, select any data series options you want, and click OK.
What is polynomial order Excel?
The order of the polynomial can be determined by the number of fluctuations in the data or by how many bends (hills and valleys) appear in the curve. An Order 2 polynomial trendline generally has
only one hill or valley. Order 3 generally has one or two hills or valleys. Order 4 generally has up to three.
How do you fit a polynomial into data points?
To perfectly fit a polynomial to data points, an order polynomial is required. To restate slightly differently, any set of points can be modeled by a polynomial of order . It can be shown that such a
polynomial exists and that there is only one polynomial that exactly fits those points.
What is polynomial trendline order?
A polynomial trendline is a curved line that is used when data fluctuates. It is useful, for example, for analyzing gains and losses over a large data set. The order of the polynomial can be
determined by the number of fluctuations in the data or by how many bends (hills and valleys) appear in the curve.
How do you use Excel to solve for a variable?
How to Use Solver in Excel
1. Click Data > Solver. You’ll see the Solver Parameters window below.
2. Set your cell objective and tell Excel your goal.
3. Choose the variable cells that Excel can change.
4. Set constraints on multiple or individual variables.
5. Once all of this information is in place, hit Solve to get your answer.
Which is the best trendline to use in Excel?
A linear trendline is a best-fit straight line that is used with simple linear data sets. Your data is linear if the pattern in its data points resembles a line. A linear trendline usually shows that
something is increasing or decreasing at a steady rate.
How do I add a second order polynomial trendline in Excel?
Graph the data using a Scatter (XY) plot in the usual way. Then click on the data to select it. From the menu choose Chart/Add Trendline…. From the window that appears, select Polynomial of Order 2.
What is second order polynomial fit?
To achieve a polynomial fit using general linear regression you must first create new workbook columns that contain the predictor (x) variable raised to powers up to the order of polynomial that you
want. For example, a second order fit requires input data of Y, x and x².
How do you use the Solve function in Excel?
For activating the solver tool we need to do the following steps:
1. Step 1: Go to File and select options.
2. Step 2: Now select the Add-ins option and click on Go and finally click on OK.
3. Step 3: After clicking OK, Select Solver Add-in and press OK.
4. Step 4: Now solver will appear in data section like this.
What is the difference between Goal Seek and Solver 10?
Goal Seek: Determines the value that you need to enter in a single input cell to produce a result that you want in a dependent (formula) cell. Solver: Determines the values that you need to enter in
multiple input cells to produce a result that you want.
Should I use linear or exponential trendline?
For example, if your chart displays a steady increase in revenue by product line over time, you could use a linear trendline. Use an exponential trendline when your data values increase or decrease
exponentially, or at an increasingly higher or lower rate.
How do you know which trendline to use?
Choose the line that fits the data best
Ideally, you should choose the trendline where all data points have the smallest distance to the trendline.
What is a 2nd order polynomial trend line?
An Order 2 polynomial trendline generally has only one hill or valley. Order 3 generally has one or two hills or valleys. Order 4 generally has up to three. The following example shows an Order 2
polynomial trendline (one hill) to illustrate the relationship between speed and gasoline consumption.
What is a 4th order polynomial?
A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form. where a ≠ 0. The derivative of a quartic function is a cubic function.
How do you fit a second-degree polynomial into data?
Can Excel solve for variables?
Excel can solve for unknown variables, either for a single cell with Goal Seeker or multiple cells with Solver. We’ll show you how it works. Excel is a powerful tool when your data is complete.
How do I find Solver in Excel?
Load the Solver Add-in in Excel
1. In Excel 2010 and later, go to File > Options.
2. Click Add-Ins, and then in the Manage box, select Excel Add-ins.
3. Click Go.
4. In the Add-Ins available box, select the Solver Add-in check box, and then click OK.
What is the purpose of Goal Seek 10?
Goal seek is an advanced spreadsheet feature that allows to provides the values for the target based input. Just decide the target value and you can select which cell should be changed in goal seek
dialog box.
Is Solver better than Goal Seek?
1) Solver can solve formulas (or equations) which use several variables whereas Goal Seek can only be used with a single variable. 2) Solver will allow you to vary the values in up to 200 cells
whereas Goal Seek only allows you to vary the value in one cell. 3) It is possible to save one (or more) models with Solver.
How do you determine the best trend line?
A trendline is most accurate when its R-squared value (R-squared value: A number from 0 to 1 that reveals how closely the estimated values for the trendline correspond to your actual data. A
trendline is most reliable when its R-squared value is at or near 1.
When should I use polynomial trendline?
How many types of trend lines are there?
There are a number of different kinds of trendlines. The most common are characterized as linear, logarithmic, polynomial, power, exponential, and moving average.
How do you find the right order for fitting polynomials?
However, it is crucial to remember that if we try to fit polynomials of a too high degree, we may overfit our model. We can find the right degree (or order) by increasing it to the point we see
enough significance to define the best possible model (called forward selection).
How to fit a polynomial curve with a certain degree in Excel?
You can use the LINEST () function in Excel to fit a polynomial curve with a certain degree. For example, you can use the following basic syntax to fit a polynomial curve with a degree of 3: The
function returns an array of coefficients that describes the polynomial fit.
How do you choose a polynomial to use?
First, you need to decide what degree of polynomial to use. Again, that is difficult to do unless you chart the data first. Suppose you choose an order-6 polynomial.
How to fit a polynomial regression equation to a data set?
Use the following steps to fit a polynomial regression equation to this dataset: Step 1: Create a scatterplot. First, we need to create a scatterplot. Go to the Charts group in the Insert tab and
click the first chart type in Scatter: Step 2: Add a trendline. Next, we need to add a trendline to the scatterplot.
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Mastering Formulas In Excel: What Is Range Formula
Mastering formulas in Excel is essential for anyone looking to become proficient in data analysis and reporting. Understanding and utilizing formulas can save time and effort, as well as improve the
accuracy of your work. One important type of formula to master is the range formula, which allows users to perform calculations on a range of cells within a worksheet.
Range formulas are versatile and can be used for a variety of purposes, from basic arithmetic operations to more complex calculations. By learning how to use range formulas effectively, Excel users
can streamline their data analysis processes and make more informed decisions based on their findings.
Key Takeaways
• Mastering formulas in Excel is essential for data analysis and reporting.
• Range formulas are versatile and can be used for basic arithmetic operations as well as more complex calculations.
• Understanding the syntax and different types of range formulas is crucial for effective usage.
• Practice and experimentation are important for mastering range formulas.
• Avoid common mistakes such as forgetting to lock cell references and overcomplicating formulas.
Understanding Range Formula
A. Definition of range formula in Excel
The range formula in Excel refers to a group of cells that are selected and used in a calculation or function. It allows users to perform operations on a collection of data within a specified range
of cells.
B. How range formulas are used in data analysis and calculations
Range formulas are extensively used in data analysis and calculations to manipulate and analyze large sets of data. They enable users to perform complex calculations, such as finding sums, averages,
and percentages, across multiple cells simultaneously.
C. Examples of common uses of range formulas
• Summing a range of values: One common use of range formulas is to easily calculate the sum of a range of values by using the SUM function. For example, =SUM(A1:A10) adds the values in cells A1
through A10.
• Calculating averages: Range formulas can also be used to find the average value of a range of cells. The AVERAGE function can be used for this purpose, such as =AVERAGE(B1:B10) to find the
average of cells B1 through B10.
• Calculating percentages: In Excel, range formulas can be used to calculate percentages of a range of values. For instance, =C2/D2 can be used to find the percentage of the value in cell C2
relative to the value in cell D2.
Syntax of Range Formula
When working with formulas in Excel, it is important to have a clear understanding of the syntax structure of each formula. The range formula is no exception and it is essential to grasp the key
components and proper formatting to ensure accurate results.
Explanation of the Syntax Structure of Range Formula
The range formula in Excel follows a specific syntax structure that consists of the following components:
• = - The equal sign indicates the beginning of the formula.
• Range - This component refers to the range of cells that the formula will operate on.
• Function - The function is the command that performs a specific operation on the range of cells.
• Arguments - These are the inputs required by the function to perform the operation.
Key Components of the Range Formula
Understanding the key components of the range formula is crucial for accurate implementation. The range formula consists of the following key components:
• Cell References - These are used to specify the range of cells that the formula will operate on.
• Mathematical Operators - These include addition (+), subtraction (-), multiplication (*), and division (/) to perform mathematical operations within the range.
• Functions - Excel offers a wide range of functions such as SUM, AVERAGE, MAX, MIN, and COUNT to perform specific calculations on the range of cells.
How to Properly Format a Range Formula in Excel
Proper formatting of the range formula is essential to ensure its accuracy and effectiveness. The following steps can be followed to format a range formula in Excel:
• Select the Cell - Click on the cell where you want the result of the formula to appear.
• Start with an Equal Sign - Begin the formula by typing an equal sign (=) in the selected cell to indicate that it is a formula.
• Specify the Range - Enter the range of cells that the formula will operate on. This can be done by typing the cell references or selecting the range with the mouse.
• Enter the Function - After specifying the range, enter the appropriate function that you want to use to perform the calculation.
• Provide Arguments - If the function requires any arguments, enter them within parentheses after the function name.
• Press Enter - Once the formula is properly formatted, press Enter to apply the formula and calculate the result.
Different Types of Range Formulas
When working with Excel, range formulas are an essential tool for performing calculations across a range of cells. There are several types of range formulas that serve different purposes.
A. SUM Range Formula: its purpose and usage
• The SUM range formula is used to add up the values in a specified range of cells. This is particularly useful when you need to quickly calculate the total of a series of numbers.
• To use the SUM range formula, simply input "=SUM(" followed by the range of cells you want to include in the calculation, separated by commas. Close the formula with a ")" and press Enter.
• For example, if you want to calculate the total sales for the month, you could use the formula "=SUM(B2:B31)" to add up the values in cells B2 to B31.
B. AVERAGE Range Formula: its purpose and usage
• The AVERAGE range formula calculates the average value of a range of cells. This is helpful when you need to find the mean value of a set of numbers.
• To use the AVERAGE range formula, enter "=AVERAGE(" followed by the range of cells you want to include in the calculation, separated by commas. Close the formula with a ")" and press Enter.
• For instance, if you want to determine the average test score for a class, you could use the formula "=AVERAGE(C2:C20)" to find the mean value of cells C2 to C20.
C. MAX and MIN Range Formulas: their purpose and usage
• The MAX range formula is used to find the highest value in a range of cells, while the MIN range formula identifies the lowest value.
• To use the MAX or MIN range formula, type "=MAX(" or "=MIN(" followed by the range of cells you want to include in the calculation, separated by commas. Close the formula with a ")" and press
• For example, if you want to determine the highest and lowest temperatures recorded in a month, you could use the formulas "=MAX(D2:D31)" and "=MIN(D2:D31)" to find the maximum and minimum values
in cells D2 to D31, respectively.
Tips for Mastering Range Formulas
Mastering range formulas in Excel can greatly improve your data analysis and reporting skills. Here are some tips to help you become proficient in using range formulas:
A. Practice using range formulas with different datasets
• 1. Start with small datasets: To build your confidence and proficiency, begin by practicing range formulas with small datasets. This will allow you to understand how the formula works and how it
can be applied to different types of data.
• 2. Gradually increase dataset complexity: As you become more comfortable with range formulas, challenge yourself by working with larger and more complex datasets. This will give you a better
understanding of how to apply range formulas in real-world scenarios.
B. Utilize online resources and tutorials for additional learning
• 1. Watch video tutorials: There are numerous online tutorials and video resources available that can help you understand and master range formulas in Excel. Take advantage of these resources to
expand your knowledge and skills.
• 2. Join online forums and communities: Engaging with Excel communities and forums can provide valuable insights and tips for using range formulas effectively. You can also seek help from
experienced users and experts in the field.
C. Experiment with combining range formulas for more complex calculations
• 1. Create complex calculations: Once you have a good grasp of individual range formulas, try combining them to create more complex calculations. This will help you understand how different
formulas can work together to produce the desired results.
• 2. Explore advanced functions: Experiment with advanced Excel functions and combine them with range formulas to perform sophisticated data analysis and calculations. This will expand your
capabilities and make you more proficient in using range formulas.
By practicing with different datasets, utilizing online resources, and experimenting with complex calculations, you can become proficient in using range formulas in Excel.
Common Mistakes to Avoid
When working with range formulas in Excel, it's important to be aware of common mistakes that can lead to errors in your calculations. By understanding these pitfalls, you can avoid them and ensure
that your formulas are accurate and reliable.
A. Forgetting to lock cell references when copying range formulas
• Issue: When you copy a range formula to other cells, Excel automatically adjusts the cell references. This can lead to incorrect results if you forget to lock the references to specific cells.
• Solution: Use the $ symbol to lock cell references when necessary. For example, if you want to keep the reference to cell A1 constant, you would use $A$1 in your formula.
B. Incorrectly specifying the range of cells in the formula
• Issue: If you specify the range of cells incorrectly in your formula, it can result in incorrect calculations.
• Solution: Double-check the range of cells you are referencing in your formula to ensure that it includes all the necessary data. This can be done by using the range selection tool or manually
inputting the cell references.
C. Overcomplicating formulas when a simpler solution would suffice
• Issue: Overcomplicating formulas can make them difficult to understand and maintain, and increases the risk of errors.
• Solution: Before finalizing a formula, consider whether there is a simpler way to achieve the same result. Break down the problem into smaller steps and use functions that are specifically
designed for the task at hand.
Mastering range formulas in Excel is crucial for effectively managing and analyzing large sets of data. Whether it's for calculating averages, finding the minimum or maximum values, or performing
complex calculations across multiple cells, range formulas are essential tools for any Excel user.
As with any skill, proficiency in using range formulas comes with practice and continuous learning. Don't be afraid to challenge yourself with new formulas or complex data sets to improve your skills
and efficiency in Excel. With dedication and practice, you'll soon be a master at using range formulas to streamline your data analysis and reporting.
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TU-BCA 1st Semester: Mathematics I - NOTE BAHADUR
Course Title: Mathematics I (3 Cr.)
Course Code: CACS104
Year/Semester: I/I
Class Load: 5 Hrs. I Week (Theory: 3 Hrs., Tutorial: 1 Hr., Practical: 1 Hr.)
Course Description
This course includes several topics from algebra and analytical geometry such as set theory and real & complex number; relation, functions and graphs; sequence and series; matrices and determinants;
permutation & combination; conic section and vector in space which are essential as mathematical foundation for computing.
Course Objectives
The general objective of this course is to provide the students with basic mathematical skills required to understand Computer Application Courses.
Course Contents
Unit 1 Set Theory and Real & Complex Number [7 Hrs.]
Concept, Notation and Specification of Sets, Types of Sets, Operations on Sets (Union, Intersection, Difference, Complement) and their Venn diagrams, Laws of Algebra of Sets (without proof), Cardinal
Number of Set and Problems Related to Sets.
Real Number System, Intervals, Absolute Value of Real Number. Introduction of Complex Number, Geometrical Representation of Complex Number, Simple Algebraic Properties of Complex Numbers (Addition,
Multiplication, Inverse, Absolute Value)
Unit 2 Relation, Functions and Graphs [8 Hrs.]
Ordered pairs, Cartesian product, Relation, Domain and Range of a relation, Inverse of a relation; Types of relations: reflective, symmetric, transitive, and equivalence relations. Definition of
function, Domain and Range of a function, Inverse function, Special functions (Identity, Constant), Algebraic (linear, Quadratic, Cubic).
Trigonometric and their graphs. Definition of exponential and logarithmic functions, Composite function, (Mathematica)
Unit 3 Sequence and Series [7 Hrs.]
Sequence and – Series (Arithmetic, Geometric, Harmonic), Properties of Arithmetic, Geometric, Harmonic sequences, A. M., G. M., and H. M. and relation among them. Sum of Infinite Geometric Series.
Taylor’s Theorem (without proof), Taylor’s series, Exponential series.
Unit 4 Matrices and Determinants [8 Hrs.]
Introductions of Matrices, Types of Matrices, Equality of Matrices, Algebra of Matrices, Determinant, Transpose, Minors and Cofactors of Matrix. Properties of determinants (without proof), Singular
and non-singular matrix, adjoin and inverse of matrices Linear transformations, orthogonal transformations; rank of matrices. (MATLAB)
Unit 5 Analytical Geometry [8 Hrs.]
Conic Sections: Definitions (Circle, Parabola, Ellipse, Hyperbola and Related Terms), Examples to Explain the Defined Terms, Equations and Graphs of The Conic Sections Defined Above, Classifying the
Defined Conic Sections by Eccentricity and Related Problems, Polar Equations of Lines, Circles, Ellipse, Parabolas, and Hyperbolas. (Mathematica / MATLAB)
Vectors in Space: Vectors in Space, Algebra of Vectors in Space, Length, Distance Between Two Points, Unit Vector, Null Vector. Scalar Product, Cross Product of Two and Three Vectors and Their
Geometrical Interpretations and Related Examples. (MATLAB)
Unit 6 Permutation and Combination [7 Hrs.]
Basic Principle of Counting, Permutation of a. Set of Objects All Different b. Set of Objects Not All Different c. Circular Arrangement d.. Repeated Use of The Same Object. Combination of Things All
Different, Properties of Combination.
Laboratory Works
Mathematica and/ or MATLAB should be used for above mentioned topics.
Teaching Methods
The general teaching pedagogy includes class lectures, group works, case studies, guest lectures, research work, project work, assignments (theoretical and practical), tutorials and examinations
(written and verbal). The teaching faculty will determine the choice of teaching pedagogy as per the need of the topics.
Text Book
1. Thomas, G. B, Finney, R. S., “Calculus with Analytic Geometry”, Addison -Wesley, 9th Edition.
Reference Books
– Bajracharya D. R., Shreshtha, R. M. & et al, “Basic Mathematics I, II” Sukunda Pustak Bhawan, Nepal
– Budnick, F. S., “Applied Mathematics for Business, Economics and the Social Sciences”, McGraw-Hill Ryerson Limited.
– Monga, G. S., “Mathematics for Management and Economics”, Vikas Publishing House Pvt. Ltd., New Delhi.
– Paudel, K. C., GC. F. B., and et. al, “Higher Secondary Mathematics”, Asmita Publication & Distributors Pvt. Ltd, Nepal.
– Upadhayay, H. P., Paudel, K.0 & ct al, “Elements of Business Mathematics”, Pinnacle Publication.
– Yamane, T. “Mathematics for Economist”, Prentice-hall of India.
To download full Syllabus CLICK HERE
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Lab and Calculations
For the circuit below, perform the following:
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Given an AC input voltage of Vrms = 200Vrms and a sinusoidal frequency of omega = 120pi rad/sec, determine the following:
V1peak (peak value of AC input voltage)
V1(peak-to-Vpeak )(peak to peak value of AC input voltage)
Frequency of input voltage V1 in Hz.
V2peak (peak value of AC output voltage)
V2(peak-to-Vpeak )(peak to peak value of AC output voltage)
Frequency of output voltage V2 in Hz.
Construct circuit in MultiSIM using calculated values for V1.
Capture a screenshot that shows V1 and V2 on an oscilloscope XSC1.
Submit calculations, MultSIM screenshot and MultiSIM circuit (.ms11) file
For the circuit below, perform the following:
Given a voltage gain of -10, determine the value of R2.
Calculate the peak-to-peak value of Vout and rms value of Vout.
Construct circuit in MultiSIM with calculated value of R2.
Capture a screenshot that shows the input voltage V1 and output voltage Vout in MultiSIM with an oscilloscope XSC1
Submit calculations, MultSIM screenshot and MultiSIM circuit (.ms11) file
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A concave mirror produces three times magnified (enlarged) real image of an object placed at 10 cm in front of it. Where is the image located?
• 3
Trayi Goyal
Asked: 2020-10-20T12:20:04+00:00 2020-10-20T12:20:04+00:00In: Class 10
A concave mirror produces three times magnified (enlarged) real image of an object placed at 10 cm in front of it. Where is the image located?
• 3
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NCERT Solution for Class 10 Science Chapter 10
Light – Reflection and Refraction
NCERT Books for Session 2022-2023
CBSE Board and UP Board
Intext Questions
Page No-171
Questions No-2
5 Answers
1. 2020-12-12T06:07:29+00:00Added an answer on December 12, 2020 at 6:07 am
Magnification produced by a spherical mirror is given by the relation,
m = Height of the image / Height of the Object = – Image distance / Object distance
m = h₁/h₀ = -v/u
Let the height of the object, ho = h
Then, height of the image, hI = −3h (Image formed is real)
-3h/h = -v/u
v/u = 3
Object distance, u = −10 cm v = 3 × (−10) = −30 cm
Here, the negative sign indicates that an inverted image is formed at a distance of 30 cm
in front of the given concave mirror.
For more answers visit to website:
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2. 2021-01-29T06:24:31+00:00Added an answer on January 29, 2021 at 6:24 am
Because the image is real, so magnification m must be negative.
m = -v/u
-3 = -v/-10
v = -30
Thus the image is located at a distance of 30 cm from the mirror on the object side of the mirror.
□ 2
□ Share
3. 2021-01-29T09:59:38+00:00Added an answer on January 29, 2021 at 9:59 am
Because the image is real, so magnification m must be negative.
m = -v/u
-3 = -v/-10
v = -30
Thus the image is located at a distance of 30 cm from the mirror.
□ 2
□ Share
4. 2021-01-30T15:39:20+00:00Added an answer on January 30, 2021 at 3:39 pm
m = − v/u= −3 or v = 3u
But u = -10
∴ v = −30cm.
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□ Share
6. 2023-11-21T07:03:09+00:00Added an answer on November 21, 2023 at 7:03 am
To find the image location in a concave mirror, you can use the mirror formula:
1/f = 1/d_0 + 1/d₁
» f is the focal length of the mirror (positive for concave mirrors),
» d_o is the object distance (distance from the object to the mirror)
» d₁ is the image distance (distance from the image to the mirror).
Given that the concave mirror produces a magnified (enlarged) real image, the magnification (m) is positive and given by the formula:
m = – d₁/d_0
In this case, you’re told that the magnification is 3, so m = 3.
Also, the object distance (d_0) is 10 cm.
Now, let’s find the image distance (d₁) using the magnification formula:
3 = −d₁/10
Solving for d₁:
d₁= − 30 cm
The negative sign indicates that the image is formed on the same side as the object (in front of the mirror). So, the real and magnified image is located 30cm in front of the concave mirror.
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Math 2 Teacher's Edition, 4th ed. | BJU Press
Math 2 Teacher's Edition, 4th ed.
Math 2 Teacher’s Edition (4th ed.) guides teachers through each lesson with background information and scaffolded questions that help assess whether students understand the concepts of the lesson. To
better help all students learn math, strategies are employed for both concepts and fact automaticity.
Christian School Pricing
Please log in to start your order.
Math 2 Teacher’s Edition (4th ed.) guides teachers through each lesson with background information and scaffolded questions that help assess whether students understand the concepts of the lesson. To
better help all students learn math, strategies are employed for both concepts and fact automaticity.
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Cameron Arnold | Prealgebra Tutor on HIX Tutor
Cameron Arnold
Sacred Heart University
Prealgebra teacher | Experienced educator in USA
I specialize in Prealgebra, holding a degree from Sacred Heart University. With a passion for simplifying complex mathematical concepts, I bring a depth of knowledge to help students grasp
foundational principles. My goal is to foster a supportive learning environment, empowering students to excel in their academic journey. Let's embark on this mathematical adventure together!
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Calculus and Analytical Geometry Lecture No. 1
Calculus and Analytical Geometry Lecture No. 1
The course contents for calculus and analytical geometry are explained below:
Limits and Continuity:
Definition: In calculus, limits and continuity are fundamental concepts. The limit of a function at a point describes its behavior as the input approaches that point. Continuity refers to the absence
of disruptions or jumps in the function’s graph.
Introduction to Functions:
Definition: A function is a relation between a set of inputs (domain) and a set of possible outputs (codomain).
Introduction to Limits:
Definition: Limits in calculus describe the behavior of a function as the input approaches a particular value. They are essential for understanding the instantaneous rate of change and continuity.
Indeterminate Forms of Limits:
Explanation: Some limits result in indeterminate forms (e.g., 0/0 or ∞/∞). Techniques like L’Hôpital’s Rule are employed to handle such cases.
Continuous and Discontinuous Functions and Their Applications:
Explanation: Continuous functions have no sudden jumps, while discontinuous functions may have disruptions. Applications include analyzing physical phenomena and modeling real-world situations.
Differential Calculus:
Definition: Differential calculus involves the study of rates of change and slopes. It includes the concept of derivatives, which measure the instantaneous rate of change of a function at a given
Concept and Idea of Differentiation:
Explanation: Differentiation involves finding the derivative of a function, representing the rate at which the function’s output changes concerning its input.
Geometrical and Physical Meaning of Derivatives:
Explanation: Geometrically, derivatives represent slopes of tangent lines. Physically, derivatives can represent velocities, accelerations, or rates of change in real-world scenarios.
Rules of Differentiation:
Explanation: Rules, such as the power rule, product rule, and chain rule, provide systematic methods for finding derivatives of various functions.
Techniques of Differentiation:
Explanation: Techniques like implicit differentiation and linear approximation offer approaches to finding derivatives in more complex situations.
Rates of Change, Tangents, and Normal Lines:
Explanation: Derivatives provide rates of change and allow the determination of tangent and normal lines to a curve at a given point.
Chain Rule, Implicit Differentiation, Linear Approximation:
Explanation: The chain rule deals with composite functions, implicit differentiation handles implicit equations, and linear approximation approximates a function using its tangent line.
Applications of Differentiation:
Explanation: Applications include finding extreme values, mean value theorems, and analyzing the behavior of functions.
Integral Calculus:
Definition: Integral calculus deals with the concept of integration, which involves finding the accumulated quantity represented by a function.
Concept and Idea of Integration:
Explanation: Integration represents the accumulation of quantities and is the reverse process of differentiation.
Indefinite Integrals:
Explanation: Indefinite integrals provide a family of antiderivatives and are written using the symbol ∫.
Techniques of Integration:
Explanation: Techniques, including substitution, integration by parts, and partial fractions, are employed to find definite and indefinite integrals.
Riemann Sums and Definite Integrals:
Explanation: Riemann sums are used to approximate definite integrals, representing the signed area under a curve.
Applications of Definite Integrals:
Explanation: Applications include finding areas under curves, calculating work done, and determining accumulated quantities.
Improper Integrals:
Explanation: Improper integrals involve integrals over unbounded intervals or functions with infinite discontinuities.
Applications of Integration:
Explanation: Applications include finding areas, and volumes, and solving real-world problems in various fields.
This summary covers the essential concepts in calculus, from limits and continuity to differentiation, integration, and their practical applications.
We invite you to continue exploring our website at www.RanaMaths.com
Calculus and Analytical Geometry Lecture 1
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Interest - (Principles of Finance) - Vocab, Definition, Explanations | Fiveable
from class:
Principles of Finance
Interest is the cost of borrowing money, typically expressed as a percentage of the principal amount. It can also be seen as the return on investment for money that is lent or invested over time.
congrats on reading the definition of interest. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Simple interest is calculated using the formula I = PRT, where P is principal, R is the rate, and T is time.
2. Compound interest involves calculating interest on both the initial principal and any accumulated interest from previous periods.
3. The future value of an investment or loan with compound interest can be found using FV = PV(1 + r/n)^(nt), where PV is present value, r is annual interest rate, n is number of compounding periods
per year, and t is time in years.
4. The concept of discounting future cash flows to present value relies heavily on understanding how interest impacts the time value of money.
5. Higher interest rates generally lead to a higher future value for savings but increase the cost of borrowing.
Review Questions
• What formula would you use to calculate simple interest?
• How does compound interest differ from simple interest?
• Why are higher interest rates significant in finance?
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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What are stochastic and deterministic processes? - The Culture SGWhat are stochastic and deterministic processes?
During class, I occasionally mention to students that the calculus you learnt is deterministic and they go uh huh. So like what is non deterministic?! I’ll attempt to give some simple definitions and
example and also touch on stochastic process.
First, a deterministic process is one that given at a particular state and an action, we only have one possible successive state to move on to. That is what we do in A-levels, since we often don’t
consider any randomness in our work.
Second, a non-deterministc process is one that given at a particular state and an action, we have a set of possible successive state to move on to.
Lastly, a stochastic process is one that to begin with, has a probability of to be at a particular state, and has a finite number of states that it can transit to. It also has transition
probabilities, which adds up to one.
A deterministic process is one where the present state completely determines the future state. If I make a (riskless) investment of
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The Frisch–Waugh–Lovell Theorem for Both OLS and 2SLS | R-bloggersThe Frisch–Waugh–Lovell Theorem for Both OLS and 2SLS
The Frisch–Waugh–Lovell Theorem for Both OLS and 2SLS
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The Frisch–Waugh–Lovell (FWL) theorem is of great practical importance for econometrics. FWL establishes that it is possible to re-specify a linear regression model in terms of orthogonal
complements. In other words, it permits econometricians to partial out right-hand-side, or control, variables. This is useful in a variety of settings. For example, there may be cases where a
researcher would like to obtain the effect and cluster-robust standard error from a model that includes many regressors, and therefore a computationally infeasible variance-covariance matrix.
Here are a number of practical examples. The first just takes a simple linear regression model, with two regressors: x1 and x2. To partial out the coefficients on the constant term and x2, we first
regress x2 on y1 and save the residuals. We then regress x2 on x1 and save the residuals. The final stage regresses the second residuals on the first. The following code illustrates how one can
obtain an identical coefficient on x1 by applying the FWL theorem.
x1 = rnorm(100)
x2 = rnorm(100)
y1 = 1 + x1 - x2 + rnorm(100)
r1 = residuals(lm(y1 ~ x2))
r2 = residuals(lm(x1 ~ x2))
# ols
coef(lm(y1 ~ x1 + x2))
# fwl ols
coef(lm(r1 ~ -1 + r2))
FWL is also relevant for all linear instrumental variable (IV) estimators. Here, I will show how this extends to the 2SLS estimator, where slightly more work is required compared to the OLS example
in the above. Here we have a matrix of instruments (Z), exogenous variables (X), and endogenous variables (Y1). Let us imagine we want the coefficient on one endogenous variable y1. In this case we
can apply FWL as follows. Regress X on each IV in Z in separate regressions, saving the residuals. Then regress X on y1, and X on y2, saving the residuals for both. In the last stage, perform a
two-stage-least-squares regression of the X on y2 residuals on the X on y2 residuals using the residuals from X on each Z as instruments. An example of this is shown in the below code.
ov = rnorm(100)
z1 = rnorm(100)
z2 = rnorm(100)
y1 = rnorm(100) + z1 + z2 + 1.5*ov
x1 = rnorm(100) + 0.5*z1 - z2
x2 = rnorm(100)
y2 = 1 + y1 - x1 + 0.3*x2 + ov + rnorm(100)
r1 = residuals(lm(z1 ~ x1 + x2))
r2 = residuals(lm(z2 ~ x1 + x2))
r3 = residuals(lm(y1 ~ x1 + x2))
r4 = residuals(lm(y2 ~ x2 + x2))
# biased coef on y1 as expected for ols
# 2sls
# fwl 2sls
The FWL can also be extended to cases where there are multiple endogenous variables. I have demonstrated this case by extending the above example to model x1 as an endogenous variable.
# 2 endogenous variables
r5 = residuals(lm(z1 ~ x2))
r6 = residuals(lm(z2 ~ x2))
r7 = residuals(lm(y1 ~ x2))
r8 = residuals(lm(x1 ~ x2))
r9 = residuals(lm(y2 ~ x2))
# 2sls coefficients
p1 = fitted.values(lm(y1~z1+z2+x2))
p2 = fitted.values(lm(x1~z1+z2+x2))
lm(y2 ~ p1 + p2 + x2)
# 2sls fwl coefficients
p3 = fitted.values(lm(r7~-1+r5+r6))
p4 = fitted.values(lm(r8~-1+r5+r6))
lm(r9 ~ p3 + p4)
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RSA algorithm and CRT in program
in Programming
RSA algorithm and CRT in program
Explained the way to compute the RSA encryption and decryption in C sharp program, and also the way to calculate of the private key provided you know the public key, P and Q by using the extended
Euclidean algorithm, reference to the Python implementation of RSA.
Testing material with RSA
For understand the using CRT with RSA, refer to this Link. And I am not going to explain the detail as I am not quite familiar with the math in detail myself.
As long as CRT will speed up the computing of RSA, here suppose we have all the detail as Public modulus, Public Exponent, P, Q, DP1, DQ1 and PQ, we use example as below, as I have tested it, it is
0x80 length of public modulus, below is the value.
Public Exponent,below is the value
0x40 – length of P, below is the value,
0x40– length of Q, below is the value,
0x40– length of DP1, below is the value,
0x40– length of DQ1, below is the value,
0x40– length of PQ, below is the value,
Private exponent(d) as below,
We use below value as the plain text to encrypt:
Encrypted text as below,
Can use this link to verify the RSA encryption and decryption result.
Verify the CRT parameters are valid
The interface of the RSA computing GUI is as below screenshot shows,
Make sure you have the correct version Visual studio to import “System.Numerics” support. as below screenshot,
To convert from hex string on the GUI to the byte array, can refer to this Link, by using below methods,
public static byte[] ConvertToByteArray(string value)
Convert all the RSA parameters to the BigIntegers,
BigInteger P = Program.FromBigEndian(P2);
BigInteger Q = Program.FromBigEndian(Q2);
BigInteger DP = Program.FromBigEndian(DP2);
BigInteger DQ = Program.FromBigEndian(DQ2);
BigInteger InverseQ = Program.FromBigEndian(InverseQ2);
BigInteger E = Program.FromBigEndian(Exponent2);
BigInteger M = Program.FromBigEndian(Modulus2);
BigInteger M1 = BigInteger.Multiply(P, Q); // M = P*Q
Verify M = P*Q
if (M1.CompareTo(M) == 0)
//MessageBox.Show("Modulus is correct!");
MessageBox.Show("Modulus is not correct as M!=P*Q");
Calculate the private key D,
BigInteger PMinus1 = BigInteger.Subtract(P, BigInteger.One); // P-1
BigInteger QMinus1 = BigInteger.Subtract(Q, BigInteger.One); // Q-1
BigInteger Phi = BigInteger.Multiply(PMinus1, QMinus1);
BigInteger D1 = Program.modinv(E, Phi);
Here the Program.modinv(E, Phi) is to use the Extended Euclidean algorithm to calculate the modular inverse, the result is the private key D, E is the public key, refer to this link. The source code
is as below,
public static BigInteger modinv(BigInteger u, BigInteger v)
BigInteger inv, u1, u3, v1, v3, t1, t3, q;
BigInteger iter;
/* Step X1. Initialise */
u1 = 1;
u3 = u;
v1 = 0;
v3 = v;
/* Remember odd/even iterations */
iter = 1;
/* Step X2. Loop while v3 != 0 */
while (v3 != 0)
/* Step X3. Divide and "Subtract" */
q = u3 / v3;
t3 = u3 % v3;
t1 = u1 + q * v1;
/* Swap */
u1 = v1; v1 = t1; u3 = v3; v3 = t3;
iter = -iter;
/* Make sure u3 = gcd(u,v) == 1 */
if (u3 != 1)
return 0; /* Error: No inverse exists */
/* Ensure a positive result */
if (iter < 0)
inv = v - u1;
inv = u1;
return inv;
After calculate the private key D, display on the GUI.
Verify the DQ,
BigInteger DQ1 = BigInteger.Remainder(D, QMinus1); // dQ = (1/e) mod (q-1)
if (DQ1.CompareTo(DQ) == 0)
//Console.WriteLine(" DQ Ok :)");
MessageBox.Show("DQ1 is not correct as dQ = (1/e) mod (q-1)");
Verify DP,
BigInteger DP1 = BigInteger.Remainder(D, PMinus1); // dP = (1/e) mod (p-1)
if (DP1.CompareTo(DP) == 0)
//Console.WriteLine(" DP Ok :)");
MessageBox.Show("DP1 is not correct as dP != (1/e) mod (p-1)");
Verify PQ,
//qInv = (1/q) mod p
BigInteger InverseQ1 = Program.modinv(Q, P);
if (InverseQ1.CompareTo(InverseQ) == 0)
//Console.WriteLine(" qInv = (1/q) mod p Ok :)");
MessageBox.Show("InverseQ1 is not correct as qInv = (1/q) mod p");
RSA encryption and decryption
By using the C# function to encrypt and decrypt, there will be length restriction, 1024 bit length key may only encrypt 117 bytes material, refer to this link (Modulus size – 11. (11 bytes is the
minimum padding possible.)). ,
byte[] encoded = rsa.Encrypt(input, false);
// byte[] encoded = rsa.Decrypt(input, false);
instead I am using the BigInteger calculation, but not sure it will work in all the case, but at least it works at my example material above,
BigInteger c = Program.FromBigEndian(input);
BigInteger m = BigInteger.ModPow(c, D, M); //decrypt
//BigInteger c = BigInteger.ModPow(m, E, M); //encrypt if it is encryption, it will like this line..
byte[] Plaintext = ConvertToByteArray(m.ToString("X"));
if (Plaintext[0] == 0x0)
byte[] newPlaintext = new byte[Plaintext.Length - 1]; //remove the first 0x00
Array.Copy(Plaintext, 1, newPlaintext, 0, Plaintext.Length - 1);
result_txt.Text = ByteArrayToHex(newPlaintext); //remove first 0
result_txt.Text = ByteArrayToHex(Plaintext);
String, Array, BigInteger convert in process
As above mentioned, to convert from hex string on the GUI to the byte array, can refer to thisLink, by using below methods,
public static byte[] ConvertToByteArray(string value)
After converted to byte array, then converted to BigInteger, here if the highest bit is 1, needs to add 0x00 at the beginning to prevent the negative value, and we can see there is Array.Reverse()
function call, it seems Microsoft BigInteger is LittleEndian,
public static BigInteger FromBigEndian(byte[] p)
if (p[p.Length - 1] > 127)
Array.Resize(ref p, p.Length + 1); //this is to prevent the negative value.
p[p.Length - 1] = 0;
return new BigInteger(p);
It is much easier to convert from BigInteger to the hex string and display on the GUI,
byte[] C = ConvertToByteArray(c.ToString("X"));
result_txt.Text = ByteArrayToHex(C);
RSA compute in Python
There is Python program available online to calculate RSA, refer to this Link, and source code download from Here. The software screenshot is as below,
2, Extended Euclidean algorithm in Wiki
4, Finding the modular inverse
5, RSACryptoServiceProvider.Encrypt Method (Byte[],Boolean)
6, Big number equation calculation
7, how-to-calculate-d-for-rsa-encryption-from-p-q-and-e
8, http://www.di-mgt.com.au/euclidean.html#extendedeuclidean
9, http://www.di-mgt.com.au/crt_rsa.html
10, Why is RSAParameters Modulus not equal product of P and Q?
11, Mapping RSA Encryption Parameters from CRT (Chinese remainder theorem) to .NET format
12, RSA: Private key calculation with Extended Euclidean Algorithm
13, RSA Encryption Problem [Size of payload data]
14, Signing a byte array of 128 bytes with RSA in C sharp
15, Bad length in RSACryptoserviceProvider
16, Converting Hex String To Corresponding Byte Array Using C#
Below references are for the Python implement RSA computing,
17, http://nmichaels.org/rsa.py
18, https://pypi.python.org/pypi/rsa#downloads
19, https://stuvel.eu/python-rsa-doc/usage.html#key-size-requirements
20, https://bitbucket.org/nmichaels/rsatool/src
21, https://pypi.python.org/pypi/pycrypto
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The aim of this tutorial is to implement the X25519 key exchange protocol in Magma, learning some useful Magma techniques along the way. X25519, defined in RFC7748, is essentially a formalization of
the techniques used in Bernstein's Curve25519 Elliptic Curve Diffie--Hellman software.
Useful references:
Montgomery arithmetic
Our first step is to define Montgomery curves and their arithmetic in Magma.
Start by downloading the Magma file Montgomery.m. This contains some useful pre-defined code, which will let you get straight to the interesting part of the tutorial more quickly.
Montgomery.m defines a function MontgomeryModel(A,B) which, given elements \(A \neq \pm 2\) and \(B \neq 0\) of some field, constructs
1. a projective plane curve \(C: BY^2Z = X(X^2 + AXZ + Z^2)\), using Magma's intrinsic Curve, which returns an object of type CrvEll;
2. the morphism \(x: C \to \mathbb{P}^1\) mapping \((X:Y:Z)\) to \((X:Z)\);
3. the Weierstrass model curve \(E: y^2 = x(x^2 + ABx + B^2)\) (as a Magma CrvEll object, created using EllipticCurve) and the isomorphism \(i: E \to C\);
4. the x-line pseudo-addition xADD operation for \(C\); and
5. the x-line pseudo-doubling xDBL operation for \(C\).
It then stores all of these associated items inside the curve \(C\) as attributes (to avoid re-computing them all the time), and returns \(C\)... At least, that's what it will do once you've finished
this exercise!
The Weierstrass model and isomorphism will be important for testing our code against the equivalent built-in operations for Weierstrass models in Magma.
Montgomery models
Our first task in completing MontgomeryModel is to construct the underlying curve \(C = C_{A,B}\). This also allows us to practice defining algebraic curves in Magma.
Start by looking for // EXERCISE 1.1 in the Montgomery.m source file, and completing the defining equation of \(C\).
Now test your code: in the Magma interpreter we should be able to replicate the following (after running load "Montgomery.m"):
> // Constructing Curve25519:
> p := 2^255 - 19;
> IsPrime(p);
> k := FiniteField(p);
> Curve25519 := MontgomeryModel(k!486662,k!1);
> Curve25519;
Curve over GF(57896044618658097711785492504343953926634992332820282019728792003\
956564819949) defined by
^3 + 5789604461865809771178549250434395392663499233282028201972879200395656\
4333287*X^2*Z + Y^2*Z + 578960446186580977117854925043439539266349923328202\
The defining equation looks a bit hairy like that, but that's what happens when you negate small quantities modulo medium-sized primes:
> -DefiningEquation(Curve25519);
X^3 + 486662*X^2*Z + 5789604461865809771178549250434395392663499233282028201972\
8792003956564819948*Y^2*Z + X*Z^2
To get the curve parameters \(A\) and \(B\) back, we can use AParameter and BParameter (defined near the top of Montgomery.m):
> // The Montgomery curve parameters
> A := AParameter(Curve25519);
> A;
> B := BParameter(Curve25519);
> B;
The map to the x-line
The arithmetic of Montgomery models is mostly computed using only \(x\)-coordinates, so our next step is to fix the projection to the \(x\)-line. This also allows us to practice defining a morphism
in Magma using the map< ... -> ... | ... > constructor.
Extend your MontgomeryModel function so that it
1. constructs the projective line over the same base field as the curve \(C\), then
2. defines the map \(x: C \to \mathbb{P}^1\) sending \((X:Y:Z) \mapsto (X:Z)\), and
3. stores \(x\) as the attribute xMap of \(C\) (we do this bit for you).
The relevant lines of Montgomery.m are
// EXERCISE 1.2: Construct PP^1 and x: C -> PP^1
PP1 := ProjectiveSpace(k,1);
/* uncomment and complete the following lines: */
// x := map< C -> PP1 | /* FILL IN IMAGE HERE */ >;
// C`xMap := x; // set associated attributed here
Note that we can access the map to the \(x\)-line using the function xMap defined near the top of Montgomery.m. Continuing the example above:
> // try out x-mapping:
> C := Curve25519; // curve defined as above
> x := xMap(C);
> pt := C![0,0,1];
> pt_PP1 := x(pt);
> pt_PP1;
(0 : 1)
Note: the \(x\) map that we have defined behaves slightly unexpectedly on the point at infinity:
> infty := C![0,1,0];
> infty;
(0 : 1 : 0)
> x(infty);
(1 : 0)
Why do you think this is?
Isomorphism to a Weierstrass model
Before going much further, we are going to need a means of effectively testing our implementation of Montgomery arithmetic. We will do this by defining an isomorphism between \(C\) and a Weierstrass
model, which is the form used by Magma's built-in elliptic curve machinery. This also lets us practice defining isomorphisms between geometric objects in Magma using the iso< ... -> ... | ... >
Extend your MontgomeryModel function so that it
1. computes an isomorphism to a Weierstrass model for \(C\), and
2. stores the isomorphism in the WeierstrassIsomorphism attribute of \(C\) (we do this bit for you).
The relevant lines of Montgomery.m are
// EXERCISE 1.3: Construct E and the isomorphism C <-> E
E := EllipticCurve(Polynomial([0,B^2,A*B,1]));
/* uncomment and fill in the following lines: */
// i := iso< E -> C | /* FILL IN IMAGES HERE */ , /* FILL IN PREIMAGES HERE */ >;
// C`WeierstrassIsomorphism := i;
For example, The isomorphism from the Weierstrass model can be accessed using WeierstrassModel:
> // Using the isomorphism to a Weierstrass model:
> C := Curve25519; // the curve above
> E, E_to_C := WeierstrassModel(C);
> infty := E_to_C(E!0);
> infty;
(0 : 1 : 0)
> // Let's build a more interesting point...
> // If E is defined by y^2 + h(x)y = f(x) for some f and h, then
> // we can recover f and h from E using HyperellipticPolynomials(E).
> f, h := HyperellipticPolynomials(E);
> assert h eq 0;
> xcoord := k!9;
> ycoord := Sqrt(Evaluate(f,xcoord));
> base_point_E := E![xcoord,ycoord];
> r := 2^252 + 27742317777372353535851937790883648493;
> IsPrime(r);
> r*base_point_E;
(0 : 1 : 0)
> base_point_C := E_to_C(base_point_E);
> base_point_C;
(9 : 14781619447589544791020593568409986887264606134616475288964881837755586237401 : 1)
The first operation to define for Montgomery curves is the pseudo-doubling operation \(x(P) \mapsto x([2]P)\). For this we will use Magma's func< ... | ... > constructor, which conveniently
constructs a function object given arguments and an expression in terms of those arguments.
Recall that if \(x(P) = (X_P:Z_P)\) and \((X_{[2]P}:Z_{[2]P}) = x([2]P)\), then \[(X_{[2]P}:Z_{[2]P}) = (Q\cdot R : S\cdot(R+\tfrac{A+2}{4}S))\] where \(Q = (X_P+Z_P)^2\), \(R = (X_P-Z_P)^2\), and \
(S = Q - R\) (so in fact \(S = 4X_PZ_P\)).
Extend your MontgomeryModel function so that it
1. constructs a function xDBL(xP), which takes one point xP = x(P) on \(\mathbb{P}^1\) and returns x2P = x([2]P) and xDBL on \(\mathbb{P}^1\), and
2. stores the result in the attribute xDBLOperation (we do this for you).
The relevant piece of code in Montgomery.m is
// EXERCISE 1.4: Pseudo-doubling
/* uncomment and fill in the following lines: */
xDBL := func< xP | /* FILL IN IMAGE HERE */ >;
C`xDBLOperation := xDBL; // set associated attribute
Hint: look at the type of what your xDBL function is supposed to be taking and returning: points on \(\mathbb{P}^1\). If PP1 is the projective line, then you can create explicit points \((a:b)\) on
it using PP1![a,b].
Hint: you can use the where keyword to break complicated expressions down into a sequence of more manageable sub-expressions. For example, the following Magma code defines a function that computes a
root of a quadratic polynomial (assuming it has one):
qroot := func< quadratic | (-b + Sqrt(Delta))/(2*a)
where Delta is (b^2 - 4*a*c)
where a is Coefficient(quadratic,2)
where b is Coefficient(quadratic,1)
where c is Coefficient(quadratic,0) >;
You can test your code as follows (this will run without any problems if your code is correct):
load "Montgomery.m";
k := GF(1009);
A := Random(k);
B := k!1;
C := MontgomeryModel(A,B);
E, E_to_C := WeierstrassModel(C);
x := xMap(C);
xDBL := xDBLOperation(C);
for trial in [1..100] do
P_E := Random(E);
P_C := E_to_C(P_E);
xP := x(P_C);
x2P := xDBL(xP);
assert x2P eq x(E_to_C(2*P_E));
end for;
(If your code is correct, then the above will run quietly without any problems...)
Now we will define pseudo-addition, which is slightly more complicated than pseudo-doubling. As before, we use Magma's func< ... | ... > constructor (though this time it will have more than one
Recall that if \(x(P) = (X_P:Z_P)\), \(x(Q) = (X_Q:Z_Q)\), \(x(P\oplus Q) = (X_\oplus: Z_\oplus)\), and \(x(P\ominus Q) = (X_\ominus: Z_\ominus)\), then \[(X_\oplus: Z_\oplus) = (Z_\ominus\cdot(U+V)^
2:X_\ominus\cdot(U-V)^2)\] where \(U = (X_P-Z_P)(X_Q+Z_Q)\) and \(V = (X_P+Z_P)(X_Q-Z_Q)\).
Extend your MontgomeryModel function so that it
1. computes a function xADD(xP,xQ,xD) which takes three points xP, xQ, and xD on \(\mathbb{P}^1\), where we assume xP \(= x(P)\), xQ \(= x(Q)\), and xD \(= x(P-Q)\) for some \(P\) and \(Q\) on \(C\)
, and returns xS \(= x(P+Q)\); and
2. stores that function in the attribute xADDOperation of \(C\) (we do this bit for you).
The relevant lines of Montgomery.m are:
// EXERCISE 1.5: Pseudo-addition
/* uncomment and fill in the following lines: */
// xADD := func< xP, xQ, xD | /* FILL IN IMAGE HERE */ >;
// C`xADDOperation := xADD; // set associated attribute
Hint: you should definitely use where here, both to simplify your code and to make the evaluation more efficient by re-using common subexpressions!
You can test your code as follows:
load "Montgomery.m";
k := GF(1009);
A := Random(k);
B := k!1;
C := MontgomeryModel(A,B);
E, E_to_C := WeierstrassModel(C);
x := xMap(C);
xADD := xADDOperation(C);
for trial in [1..100] do
P_E := Random(E);
Q_E := Random(E);
xP := x(E_to_C(P_E));
xQ := x(E_to_C(Q_E));
if xP eq xQ then // we should use xDBL, not xADD, in this case
end if;
xD := x(E_to_C(P_E - Q_E));
xS := xADD(xP,xQ,xD);
assert xS eq x(E_to_C(P_E + Q_E));
end for;
(If your code is correct, this should run quietly without any problems.)
Scalar multiplication on Montgomery models
Now we can implement scalar multiplication for Montgomery curves \(C\). Our basic tool is the Montgomery ladder, which implements the pseudoscalar multiplication on \(\mathbb{P}^1\) associated with \
(C\). We can extend this to full scalar multiplication on \(C\) using Okeya and Sakurai's \(y\)-coordinate recovery trick.
The Montgomery ladder
First, recall Montgomery's ladder algorithm (here in Python):
def ladder(m,xP):
# The Montgomery laddder: compute x([m]P) given x(P) and m
# Assumes m is positive
reg_0 = [1,0] # Image on PP^1 of pt at infinity
reg_1 = xP
for b in bits(m): # from most significant down to least significant bit
if b == 0:
(reg_0,reg_1) = (xDBL(reg_0),xADD(reg_0,reg_1,xP))
(reg_0,reg_1) = (xADD(reg_0,reg_1,xP),xDBL(reg_1))
end for
return reg_0
Define a new function Ladder(C,m,xP) implementing this algorithm in Montgomery.m. Your function should take a Montgomery model \(C\), a positive integer \(m\), and a point xP \(= x(P)\) in \(\mathbb
{P}^1(k)\) for some \(P\) in \(C(k)\), and return the point \(x([m]P)\) in \(\mathbb{P}^1(k)\).
Hint: to get a sequence containing the bits of \(m\) in the right order, from most to least significant, you can use Reverse(IntegerToSequence(m,2)).
Test your code using the following examples:
> C := Curve25519; // curve defined above
> x := xMap(C);
> P := C![9,Sqrt(k!9*(9^2 + 486662*9 + 1)),1]; // base point
> x(P);
(9 : 1)
> Ladder(C,1,x(P));
(9 : 1)
> Ladder(C,2,x(P));
(14847277145635483483963372537557091634710985132825781088887140890597596352251 : 1)
> Ladder(C,101,x(P));
(15872060397774487147062612687452767107535139408010832422811284850336798704295 : 1)
> r := 2^252 + 27742317777372353535851937790883648493;
> Ladder(C,r-1,x(P));
(9 : 1)
> Ladder(C,r,x(P));
(1 : 0)
You can also generate random examples as follows:
C := Curve25519; // as above
E, E_to_C := WeierstrassModel(C);
for trial in [1..1000] do
pt_E := Random(E);
m := Random(4*r);
assert Ladder(C,m,x(E_to_C(pt_E))) eq x(E_to_C(m*pt_E));
end for;
(If your code is correct, this should run quietly without any problems.)
Optional: y-coordinate recovery
Note: This part is not required for the key exchange implementation later, so you can skip over it and come back later if you get stuck.
Recall that the Montgomery ladder maintains two "register" variables: the first finally contains \(x([m]P)\), while the second contains \(x([m+1]P)\). Okeya and Sakurai, following an idea of Lopéz
and Dahab, showed that if \(P = (x_P:y_P:1)\) and \(Q = (x_Q:y_Q:1)\) are points on \(C\) with \(y_P \neq 0\neq y_Q\), and \(P + Q = (x_{P+Q}:y_{P+Q}:1)\), then \[2B\cdot y_P\cdot y_Q = (x_Px_Q + 1)
(x_P+x_Q + 2A) - 2A - (x_P - x_Q)^2\cdot x_{P+Q}.\] Note that \(y_{P+Q}\) never appears in this formula! The idea is to use this formula to recover \(y_Q\) given \(x_P\), \(y_P\), \(x_Q\), and \(x_
{P+Q}\). (We are not really interested in the special case where \(y_P = 0\) or \(y_Q = 0\).)
If we take \(Q = [m]P\) in the above, then we can use it to recover \(y_Q = y_{[m]P}\), and hence the full point \([m]P\), given only \(P\), \(x([m]P)\), and \(x([m+1]P)\).
1. extend your Ladder function to return the value of both of its "register" variables; and
2. define a new function ScalarMultiply for \(C\) that computes scalar multiplication on the curve \(C\) using the new version of Ladder as a subroutine.
We suggest something like this:
function ScalarMultiply(C,m,P)
// Computes [m]P on the Montgomery model C.
// Assumes m is positive.
// 1. Compute xP = x(P)
/* complete here */
// 2. Apply the ladder
xmP, xm_plus_1P := Ladder(C,m,xP);
// 3. Recover the correct preimage of [m]P:
X_mP := xmP[1];
Z_mP := xmP[2];
Y_mP := /* complete here */ ;
return C![X_mP,Y_mP,Z_mP];
end function;
Test your code:
C := Curve25519; // as above
E, E_to_C := WeierstrassModel(C);
for trial in [1..1000] do
pt_E := Random(E);
m := Random(4*r);
assert ScalarMultiply(C,m,E_to_C(pt_E)) eq E_to_C(m*pt_E);
end for;
(If your code is correct, this should run quietly without any problems.)
Branch-free code
For cryptographic applications, it is important that the runtime execution of algorithms is always independent of secret values (otherwise we are even more vulnerable to simple side-channel attacks).
In particular,
• we cannot have any branching (if statements) on bits of secrets, and
• we cannot have any array indexing (or memory access) based on bits of secrets.
In the context of Diffie-Hellman, the secrets are the scalars that we use in calls to Ladder, so we must remove the if statement from the main loop (which branches depending on bits of the scalar).
1. Write a conditional-swap function CSWAP(bit,val_0,val_1) which returns val_0,val_1 if bit is 0 and val_1,val_0 if bit is 1.
2. Now replace the if statement in your Ladder function with calls to CSWAP to add a layer of side-channel protection to your code.
Don't forget to test your code using the same examples as in Exercise 2.1 above.
X25519 key exchange
Now that we have got a working version of Montgomery arithmetic, we can use it to implement a serious cryptographic key-exchange function: X25519, which is defined in RFC7748.
Before going any further, download the file X25519.m, which contains some more skeleton code.
Compression and decompression
So far we have been applying the ladder to elements of \(\mathbb{P}^1(k)\). For key exchange we want to avoid the ambiguity (and extra space requirements) of projective points by mapping each to a
single field element, which we then represent as a 255-bit integer (ie, a 32-byte value with the most significant bit masked to 0).
In X25519 we follow Bernstein's suggestion:
• "affine" points \((X:Z)\) with \(Z \neq 0\) map to \(X/Z\);
• the point "at infinity", \((X:0)\), maps to \(0\).
If we are working over \(\mathbb{F}_p\), then we can compute all of this in "constant time" via \((X:Z) \mapsto XZ^{(p-2)}\) in \(\mathbb{F}_p\); we can then get a 256-bit integer representation
using Integers( )!value, where value is the result of the computation in \(\mathbb{F}_p\).
Going in the opposite direction, we map any element \(x\) of \(\mathbb{F}_p\) to the point \((x:1)\) in \(\mathbb{P}^1(\mathbb{F}_p)\).
Open X25519.m and complete the functions
• Compressed, mapping points in \(\mathbb{P}^1(\mathbb{F}_p)\) to 256-bit integers as above,
• Decompressed, mapping 256-bit integers to elements of \(\mathbb{F}_p\) and then to \(\mathbb{P}^1(\mathbb{F}_p)\) as above.
Scalar clamping
The scalars used in X25519 have some special properties to ensure that the scalar multiplication algorithm runs in a regular way, and to avoid some possible attacks involving the use of small-order
To convert a random integer to an X25519 scalar, we use a procedure that has become known as clamping:
1. the value is truncated (or filled with zeroes if necessary) to a 256-bit value \(x_0x_1\ldots x_{255}\);
2. bit 255 is masked to 0;
3. bit 254 is set to 1;
4. bits 0, 1, and 2 are masked to 0.
This means that the set of all X25519 scalars is \(\{ 2^{254} + 8x : x \in [0,2^{251}) \}\).
Complete the function Clamped(x) in X25519.m, which takes any integer \(x\) and returns its "clamped" value as defined above.
Hint: in Magma, this will be much easier to do using integer operations than by working on individual bits.
The X25519 function
You can now define the X25519 function, ready to carry out key exchange:
function X25519(m,u)
// The X25519 function, as defined in RFC7748.
// The arguments m and u are non-negative integers
// to be interpreted as 256-bit values (truncating if necessary).
// Step 1: Clamp m
m := Clamped(m);
// Step 2: Decompress the point
xP := Decompressed(u);
// Step 3: Apply the ladder
xmP := Ladder(Curve25519,m,xP);
// Step 4: Compress the result
v := Compressed(xmP);
return v;
end function;
We are going to test your code using the test vectors from RFC7748.
> test_scalar := 0xc49a44ba44226a50185afcc10a4c1462dd5e46824b15163b9d7c52f06be346a5
> test_base := 0x4c1cabd0a603a9103b35b326ec2466727c5fb124a4c19435db3030586768dbe6
> expected := 0x5285a2775507b454f7711c4903cfec324f088df24dea948e90c6e99d3755dac3
> X25519(test_scalar,test_base) eq expected;
The second kind of test involves iterating X25519 calls.
function iterated_test(n)
m := 9;
u := 9;
for i in [1..n] do
new_m := X25519(m,u);
u := m;
m := new_m;
end for;
return m;
end function;
What should happen is this:
> expected_1 := 0x7930ae1103e8603c784b85b67bb897789f27b72b3e0b35a1bcd727627a8e2c42;
> iterated_test(1) eq expected_1;
> expected_1000 := 0x684cf59ba83309552800ef566f2f4d3c1c3887c49360e3875f2eb94d99532c51;
> iterated_test(1000) eq expected_1000;
There is a further standard test with 1000000 iterations, but given the overhead of the Magma function calls, this will take a long time to run! But if you have an hour or two to spare and some
cycles to burn, then you can check that iterated_test(1000000) eq 0x2454664fd2d24d5fdf303c88c001c63b6f5e577e29974486fd8625abe011397c.
Key exchange
To complete a Diffie-Hellman key exchange using X25519, we proceed as follows. First, we work in the subgroup of Curve25519 generated by a point with \(x\)-coordinate 9: that is, \(9\) is the
compressed form of the public "generator".
• Alice chooses her secret \(a\), a 32-byte (256-bit) value, and computes his public key K_A = X25519(a,9), which he then publishes.
• Bob chooses his secret \(b\), a 32-byte (256-bit) value, and computes his public key K_B = X25519(b,9).
• Alice computes K = X25519(a,K_B) and Bob computes K = X25519(b,K_A); the result is their shared secret.
• To convert K into a cryptographic secret key, Alice and Bob should both apply a key derivation function to \(K\), \(K_A\), and \(K_B\).
Check your code by simulating this example:
Alice's private key, a:
Alice's public key, X25519(a, 9):
Bob's private key, b:
Bob's public key, X25519(b, 9):
Their shared secret, K:
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Linear Least Squares
Last updated:
Fitting a 2D ellipse from a set of points can be accomplished by least squares.
Least squares models are ubiquitous in science and engineering. In fact, the term least squares can have various meanings in different contexts:
• Algebraically, it is a procedure to find an approximate solution of an overdetermined linear system — instead of trying to solve the equations exactly, we minimize the sum of the squares of the
• In statistics, the least squares method has important interpretations. For example, under certain assumptions about the data, it can be shown that the estimate coincides with the
maximum-likelihood estimate of the parameters of a linear statistical model. Least square estimates are also important even when they do not coincide exactly with a maximum-likelihood estimate.
• Computationally, linear least squares problems are usually solved by means of certain orthogonal matrix factorizations such as QR and SVD.
Computing Least Square Solutions
Given a matrix $A \in \mathbf{R}^{n \times m}$ and a right-hand side vector $b \in \mathbf{R}^{n}$ with $m < n$, we consider the minimization problem
$\begin{equation} \tilde{x} = \textrm{arg min}_{x} \\| Ax - b \\|_2 \end{equation}$
Solving the Normal Equations
As most linear algebra textbooks show, the most straightforward method to compute a least squares solution is to solve the normal equations
$\begin{equation} A^t A x = A^t b \end{equation}$
This procedure can be implemented in Julia as:
x = A^t A \ A^t b
However, this is not the most convenient method from a numerical viewpoint, as the matrix $A^t A$ has a condition number of approximately $cond(A)^2$, and can therefore lead to amplification of
errors. The QR method provides a much better alternative.
Least squares via QR Decomposition
Another way of solving the Least Squares problem is by means of the QR decomposition (see Wiki), which decomposes a given matrix into the product of an orthogonal matrix Q and an upper-triangular
matrix R .
Replacing $A = QR$, the normal equations now read:
$\begin{equation} (QR)^t QR x = (QR)^t b \end{equation}$
Now, given the identity $Q^tQ = I$, this expression can be simplified to
$\begin{equation} R x = Q^t b, \end{equation}$
For well-behaved problems, at this point we have a linear system with a unique solution. As the matrix $R$ is upper-triangular, we can solve for $x$ by back-substitution.
In Julia, the least-squares solution by the QR method is built-in the backslash operator, so if we are interested in obtaining the solution of an overdetermined linear system for a given right-hand
side, we simply need to do:
x = A\b
There are cases where we want to obtain and store the matrices Q and R from the factorization. An example is when, for performance, we want to pre-compute these matrices and reuse them for multiple
right-hand sides. In such case, we do
qrA = qr(A); # QR decomposition
x = qrA\b;
Example: Fitting a 2D Ellipse from a Set of Points
Let’s focus in an interesting curve-fitting problem, where we are given $n$ pairs of points $x_i, y_i$ and we want to find the ellipse which provides the best fit.
There are indeed many ways of solving this curve-fitting problem (see for example this paper.
The most basic method starts with noting that the points of a general conic satisfy the following implicit equation:
$\begin{equation} a x^2 + bxy + c y^2 + dx + ey + f = 0 \end{equation}$
The above equation can be normalized in various ways. For example, we can resort to the alternative formulation:
$\begin{equation} a x^2 + bxy + c y^2 + dx + ey = 1 \end{equation}$
This leads to the problem of finding the best-fitting coefficients $a,b,c,d,e$ as a least-squares problem for the following $n$ equations with 5 unknowns:
$\begin{equation} a x_i^2 + b x_i y_i + c y_i^2 + dx_i + ey_i = 1 \qquad 1 \le i \le n \end{equation}$
Let’s generate some random test data
θ = π/7; a = 2; b = 1.5; x_0 = 3; y_0 = -1;
fx(t) = a*cos(θ)*cos(t) - b*sin(θ)*sin(t) + x_0
fy(t) = a*sin(θ)*sin(t) + b*cos(θ)*cos(t) + y_0
N = 200;
ts = LinRange(0,2π,N);
x = fx.(ts) + randn(N)*0.1;
y = fy.(ts) + randn(N)*0.1;
We now construct the design matrix
A = [x.^2 y.^2 x.*y x y ]
and compute the least-squares solution with the backslash operator:
p = A\ones(N)
Thats it! We have obtained the parameters of the implicit equation.
Plotting the Implicit Curve as a Countour Line
However, if we want to plot the resulting curve, we must solve an additional problem. Given $a,b,c,d,e$, how can we find points satisfying:
$\begin{equation} a x^2 + bxy + c y^2 + dx + ey = 1 \end{equation}$
and plot them?
Fortunately, this can be easily accomplished by resorting to contour plots. In order to do this, we construct a grid and we evaluate the (bivariate) polynomial expression field. Then, we seek the
contour level equal to one. Under the hood, the plotting code is using the marching squares algoritm.
using Plots
X = LinRange(minimum(x),maximum(x),100)
Y = LinRange(minimum(y),maximum(y),100)
F = Array{Float64}(undef,100,100)
for i ∈ 1:100, j ∈ 1:100
F[i,j] = p[1]*X[i]^2 + p[2]*Y[j]^2 + p[3]*X[i]*Y[j] + p[4]*X[i] + p[5]*Y[j]
plot(x,y,seriestype = :scatter, label="Observations", legend=:topleft)
plot!(fx.(ts), fy.(ts), linewidth=3, label="True Ellipse")
contour!(X, Y, F, linewidth=3, levels=[1], color=:green)
plot!([], color=:green, label="Fitted Ellipse")
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Bootstrap Sweep Circuit - Electronic Circuits and Pulse Circuits Lab
Bootstrap Sweep Circuit
1. Study the operation and working principle of Boot-strap Sweep Circuit.
2. Study the procedure for conducting the experiment in the lab.
1. To design a Boot-strap Sweep Circuit.
2. To obtain a sweep wave form.
Components Required:
1. Resistors – 100kohms , 5.6kohms, 10 Kohms - 1 each
2. Capacitors – 0.1microF, 10microF, 100microF - 1 each
3. IN4007 Diode – 1 No.
4. 2N2369 Transistors – 2 Nos.
5. Bread Board
Apparatus Required:
1. Power supply (0V-30V)
2. CRO(1Hz-20MHz)
3. Signal generator (1Hz-1MHZ)
4. Connecting Wires.
Circuit Diagram:
The input to Q[1] is the gating waveform. Before the application of the gating waveform, at t = 0, transistor Q[1] is in saturation. The voltage across the capacitor C and at the base of Q[2] is V[CE
(sat)]. To ensure Q[1] to be in saturation for t = 0, it is necessary that its current be at least equal to I[CE] / h[FE] so that R[b] < h[fe]R.
With the application of the gating waveform at t = 0, Q[1] is driven OFF. The current I[C1] now flow into C and assuming unity gain in the emitter follower V[0]. When the sweep starts, the diode is
reverse biased, as already explained above, the current through R is supplied by C[1]. The current V[CC] / R through C and R now flows from base to emitter of Q[2].if the output V[0] reaches the
voltage V[CC] in a time T[S] / Tg, then from above we have T[S] = RC.
• Connect the circuit as shown in figure.
• Apply the square wave or rectangular wave form at the input terminals.
• Connect the CRO at output terminals now plug the power card into line switch on and observe the power indication.
• As mentioned in circuit practical calculation. Observe and record the output waveforms from CRO and compare with theoretical values.
Expected Waveforms:
Inference: Conclusions can be made on sweep time T[S] and retrace time T[R] and sweep voltage V[S] of the sweep waveform theoretically and practically and also made on if the output waveform of the
Bootstrap are identical with the theoretical wave forms or not.
Viva Questions:
1. Define (a) Voltage time base generator, (b) current time base generator (c) linear time base generator.
1. A voltage time-base generator is one that provides an output voltage waveform, a portion of which exhibits a linear variation with time.
2. A current time-base generator is one that provides an output current wave form, a portion of which exhibits a linear variation with time.
3. A linear time-base generator is one that provides an output waveform a portion of which exhibits a linear variation of voltage or current with time.
2. What is the relation between the slope error, displacement error and transmission error?
Ans: The relation between slope, displacement and transmission is given as es = 2et = 8ed.
3. What are the various methods of generating time base wave-form?
Ans: The methods of generating a time-base waveform are exponential charging, constant current charging, the miller circuit, the phantastron circuit, the bootstrap circuit, compensating networks, an
inductor circuit.
4. Which amplifier is used in Boot-strap time base generator?
Ans: In bootstrap time-base generator, a non inverting amplifier with unity gain is required.
5. Which type of sweep does a bootstrap time-base generator produce?
Ans: A bootstrap time-base generator produces a positive-going sweep.
6. What is the gain of the amplifier used in Bootstrap time base generator?
Ans: In bootstrap time-base generator, a non inverting amplifier with unity gain is required.
7. What is retrace time? Write the formula for the same for Bootstrap time base generator.
Ans: The time taken by signal to return to its initial value is called retrace time. The retrace time is given by Tr = (CV[S] / V[CC])/(h[FE]/R[B]) - (1/R).
8. What is the formula for sweep amplitude in Bootstrap time base generator?
Ans: The formula for sweep amplitude in bootstrap time base generator is given as V[S] = V[CC]Tg / RC.
9. To have less flatness time of sweep signal, then the gate signal time has to be __.
Ans: To have less flatness time of sweep signal, then gate signal(Tg) has to be equivalent to sweep time (Ts).
10. A Bootstrap sweep circuit employs __ type of feedback.
Ans: A bootstrap sweep circuit employs positive feedback.
Design problem:
1. Design Boot-strap Sweep Circuit with sweep amplitude of 8V, with sweep interval of 1ms neglect flyback time and e[s ]= 0.25.
2. Design Boot-strap Sweep Circuit with sweep amplitude of 15V, with sweep interval of 2ms neglect flyback time and e[s] = 0.1.
Outcomes: After finishing this experiment students are able to Design Boot-strap sweep circuit and able to generate a sweep voltage waveform.
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Learn Excel 365 Expert Skills with The Smart Method - TheSmartMethod.com
ISBN: 978-1909253506
Dimensions: 8.27 x 1.48 x 11.69 inches
Weight: 4.17 pounds
The Data Insights feature is one of Excel 365’s killer features that will make users want to upgrade from Excel 2021.
The book will show you how to use the new Data Insights feature to work faster and smarter with Excel 365. The above Pivot Chart was created in two clicks and has spotted that two customers are
purchasing far more than others.
Natural Language Queries are a new feature for the Jan 2022 sixth edition. This is one of Excel 365’s most impressive features. You can ask questions about your data just by typing in a plain English
The book will teach you to leverage upon this amazing new feature to produce complex analysis results in seconds.
Custom Data Types are a new feature for the Jan 2022 sixth edition. Excel now has the ability to create objects with multiple properties that you can use directly in your worksheets.
The book will teach you how to use custom data types to create the multi-lingual form shown above without using any formulas.
The core idea behind Power Maps (also known as 3D Maps) is to show visualizations of Excel data upon the surface of a map but this doesn’t begin to describe the huge feature set of this powerful
Excel tool.
The book covers Power Maps in depth and that includes geocoding, data cards, multiple layers, annotations, scenes and fly-pasts. You’ll even create a HD video as you fly over your data.
The above form was created in a few minutes using Excel’s simple-to-use Form Controls.
The book will teach you how to add Option Buttons, Check Boxes, Combo Boxes, Spin Buttons, Buttons and many other controls. Once added they are really easy to bring to life. The screen grab above
shows a Mortgage Calculator that you’ll create in one of the book’s sessions.
Fuzzy Logic gives Excel the ability to match data even when the match isn’t perfect.
The book will teach you to perform Fuzzy Logic using Jaccard similarity. All of the above mis-spellings were automatically matched perfectly. This feature could save you many hours of work when
working with external data.
It is often said that the ability to fully understand Pivot Tables separates competent Excel users from Excel novices. The above grab shows two pivot tables and two slicers combined into a dashboard.
The book will teach you absolutely everything there is to know about Pivot Tables. including slicers, calculated fields, calculated items, grouping, filter fields and multiple summations.
Excel’s new Dynamic Arrays feature has changed best-practice for many common business problems.
The book will teach you all of the new Dynamic Array functions: XLOOKUP (the modern replacement for VOOKUP), XMATCH, UNIQUE, SORT, SORTBY, FILTER, RANDARRY and SEQUENCE. You’ll also learn to use
Spilled Arrays.
Excel Experts don’t use old-fashioned A1 style references when referencing dynamic data. They use tables and modern structured table references.
The book will teach you the professional way to work with tables and structured references and not ranges and A1 style references.
Excel only supports a million rows, but when Excel teams with Power Pivot, Excel can analyse billions of rows of data in the blink of an eye using a Power Pivot Data Model.
The book will teach you to create the special Star Schema data models required by Excel using Power Query and Power Pivot. This new way of working is often called Modern Data Analysis.
This isn’t an ordinary Excel pivot table – it’s an Excel Power Pivot Table (also called an OLAP Pivot Table). You can see that it can analyse multiple related tables.
The book will teach you how to prepare the relational data models needed to work with Power Pivot tables. You’ll then have the power to work faster and smarter than ever before.
DAX functions are very different to Excel functions but you’ll find many that are named just like their Excel counterparts.
The book will teach you important DAX concepts including calculated columns, calculated measures, implicit measures, explicit measures, row and filter context and the DISTINCOUNT , CALCULATE and ALL
7 Responses
1. Hi. Mac air can’t open .exe files how can I open?
1. You are on the product page for the Windows 365 Expert Skills book. Click Sample Files on the top menu and you’ll find the Apple Mac sample files there.
2. When will Excel 365 Expert Skills January 2022 version be available?
1. Excel 2021 Expert Skills published yesterday (3rd Feb 2022). Really hoping to have the sixth edition of 365 published (in paper form at least) by 15th February. The sixth edition will is
being written against the January 2201 current version. The January 2021 fifth edition is still very current and if you need to start learning quickly it is still far more up to date than any
other book.
2. Excel 365 Expert Skills 6th Edition available right now in both paper and e-Book format! Published on 7th February 2022 using the very latest January 2002 version 2201. There are some great
new features in the January 2022 release (see above) but my favourite is Custom Data Types.
3. Can I use the single ebook version on 2 of my pc’s ?
1. Yes you can.
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One Dimensional Rational Cubic Bezier Curve
Click and drag control points to change curve. Modify weights in boxes below the curve.
For more information, check out the post on my blog: Bezier Curves.
W1: W2: W3: W4:
Rational cubic bezier curves have 4 control points, a weight per control point (4 total), and total up the values of the 4 functions below to get the final point at time t.
1. A * W1 * (1-t)^3
2. B * W2 * 3t(1-t)^2
3. C * W3 * 3t^2(1-t)
4. D * W4 * t^3
They then divide that by the total of these 4 functions.
1. W1 * (1-t)^3
2. W2 * 3t(1-t)^2
3. W3 * 3t^2(1-t)
4. W4 * t^3
t - "time", but in our case we are going to use the x axis value for t.
A - The first control point, which is also the value of the function when x = 0.
B - The second control point.
C - The third control point.
D - The fourth control point, which is also the value of the function when x = 1.
W1 - The weighting of control point A.
W2 - The weighting of control point B.
W3 - The weighting of control point C.
W4 - The weighting of control point D.
In this particular case, A, B, C and D are scalars, which makes the curve into the function:
y = (A * W1 * (1-x)^3 + B * W2 * 3x(1-x)^2 + C * W3 * 3x^2(1-x) + D * W4 * x^3) / (W1 * (1-x)^3 + W2 * 3x(1-x)^2 + W3 * 3x^2(1-x) + W4 * x^3)
Note that this bezier curve is 1 dimensional because A,B,C,D are 1 dimensional, but you could use these same equations in any dimension. Also, these control points range from 0 to 1 on the X axis,
but you could scale the X axis and/or the Y axis to get a different range of values.
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PYRATZLABS Community - Mi-cho-coq tutorial • A quick introduction to Coq
In this new series of blog posts, we will be introducing Coq and Mi-Cho-Coq for proving Tezos smart contracts written in or compiled to Michelson. This series is structured as follows:
• 01. Formal proving: What is it about, what are Coq and Mi-Cho-Coq, and why will we need them.
• 02. A quick introduction to Coq
• 03. A quick introduction to Mi-Cho-Coq: proving our first Michelson smart contract.
• 04. Proving the increment contract
• 05. Proving if conditions.
• 06. Proving a token minting entrypoint
Chapter II
The context of this series on proving formally Michelson smart contracts having now been introduced, we can get started using Coq. We’ll first setup a working environment before introducing some of
Coq’s core concepts. Please note that this chapter is heavily inspired by the Software Foundations’ first chapter on functional programming in Coq. It is much lighter, but provides many of the
examples provided in Software Foundations as I found them very instructive. The explanations are ours however, as to provide a new insight, and we therefore strongly encourage our readers to go back
to the Software Foundations’ first chapter after reading this article to get a deeper understanding of the concepts introduced here.
Getting started: launching the Coq IDE
The simplest way to get started using Coq is through the CoqIDE, which we provide a Docker image for:
We can now hop into the container and launch the Coq IDE with some file:
The Coq language
In Coq, we define everything we’ll need to prove some proposition. Let’s define the boolean type, which take two values:
Easy! We can now define some functions operating on booleans. You’ll notice that Coq’s language is syntactically very similar to Ocaml’s, here’s how to implement a function negates its boolean input
using pattern matching in Coq:
Similarly, we can define the and and or functions:
Which we can evaluate as follows:
Let’s now implement some tests for these functions:
Here, we’re proving test_orb1 as follows:
1. We simplify both side using the simpl tactic, which results in true = true
2. We then end the proof by asserting that both sides are equal using the reflexivity tactic.
Of course, we don’t have to break lines between tactics:
Coq also lets us modify its reader to introduce useful new notations:
We can also define nested types, for example:
And functions operating on them:
Natural numbers
Let’s now have some fun with natural numbers. We’ll define them in Coq as follows:
So a natural number is either O, representing 0, or the successor of any number n. This is an example of a recursive type. The number 0 is associated with O, 1 with S(O), 2 with S(S(O)), etc.
We can now easily define the predecessor function:
The predecessor of S(S(O)) is S(O), which is the successor of 0, which is obviously 1.
We can use recursive functions to define the plus and multiplication operations on natural numbers as follows:
Note that just as Ocaml required the keyword rec when defining a recursive function, Coq requires the Fixpoint one. Let’s prove that forall n in natural numbers, n + 0 = n:
Here things becomes interesting, we’ve used the keyword forall which allow us to prove our proposition of all the values of n allowed by its type. We use intros n to move n from the quantifier in the
goal to the context of the current assumption. If you step through this code in the Coq IDE, you should see the following:
Let’s work through a more complex example. Here we want to prove that forall n, m in the natural numbers, the hypothesis n = m implies that n + n = m + m. We use the -> symbol between the hypothesis
and its implication:
What we’ve done is introduced the hypothesis in the context of the current assumption and substituted it in the goal. Applying reflexivity then ends the proof.
Finally, it is now time to introduce case analysis. Say that we want to prove that forall b, c in booleans, b and c = c and b, we’ll need to essentially show that this holds for all four possible
values of b and c. We can do this by using the destruct tactic. This tactic takes a variable that was introduced in the assumption context as argument, and, for a boolean, will create two subgoals:
one where a = true, and one where a = false. Hence, by destructing both a and b and applying the reflexivity tactic, we can easily prove the following theorem:
This concludes our introduction to Coq. We have seen some of the most basic tactics, and will learn many more when we will encounter them while proving Michelson contracts. This chapter should have
given you some rudimentary material to explore Coq in more detail by yourself by reading Software Foundations’ first chapter on functional programming in Coq, as well as further chapters from the
same book.
In the next chapter, we will get started with Mi-Cho-Coq by proving the identity contract that stores outputs a new storage equal to its input.
Go to the next part :
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Differentialcalculusbygorakhprasadpdfdownload Differential C
Differential Calculus by Gorakh Prasad PDF Download
Differential calculus is a branch of mathematics that studies the rates of change of functions and their properties. It is one of the core topics in calculus, along with integral calculus and
analytical geometry. Differential calculus has many applications in science, engineering, economics, and other fields.
One of the classic textbooks on differential calculus is the one written by Dr. Gorakh Prasad, a renowned Indian mathematician and educator. He was born in 1872 in Benares (now Varanasi) and obtained
his PhD from the University of Calcutta in 1904. He was a professor of mathematics at Queen's College, Benares, and later at Banaras Hindu University. He wrote several books on mathematics, including
Text Book on Differential Calculus, Text Book on Integral Calculus, Text Book on Analytical Geometry, and Text Book on Algebra.
Download File: https://urlca.com/2w3nld
The Text Book on Differential Calculus by Gorakh Prasad was first published in 1938 by the Henares Mathematical Society, Benares. It covers the topics of functions, limits, continuity, derivatives,
differentials, maxima and minima, mean value theorem, Taylor's theorem, indeterminate forms, curvature, asymptotes, envelopes, evolutes, and applications of differential calculus to geometry and
physics. The book contains numerous examples and exercises to illustrate the concepts and techniques of differential calculus. The book is suitable for students of intermediate and advanced level
The Text Book on Differential Calculus by Gorakh Prasad is available for free download from the Internet Archive . The PDF files are scanned copies of the original editions of the book. The quality
of the scans may vary depending on the source. The files are also searchable and can be viewed online or downloaded for offline reading. The book is in the public domain and can be used for personal
or educational purposes.
If you are looking for a comprehensive and classic textbook on differential calculus, you may want to check out the Text Book on Differential Calculus by Gorakh Prasad PDF download from the Internet
Archive . You will find a wealth of knowledge and insight from one of the eminent mathematicians of India.
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ch C
723 research outputs found
We study, using direct orbit integrations, the kinematic response of the outer stellar disk to the presence of a central bar, as in the Milky-Way. We find that the bar's outer Lindblad resonance
(OLR) causes significant perturbations of the velocity moments. With increasing velocity dispersion, the radius of these perturbations is shifted outwards, beyond the nominal position of the OLR, but
also the disk becomes less responsive. If we follow Dehnen (2000) in assuming that the OLR occurs just inside the Solar circle and that the Sun lags the bar major axis by ~20 degrees, we find (1) no
significant radial motion of the local standard of rest (LSR), (2) a vertex deviation of \~10 degrees and (3) a lower ratio sigma_2/sigma_1 of the principal components of the velocity- dispersion
tensor than for an unperturbed disk. All of these are actually consistent with the observations of the Solar-neighbourhood kinematics. Thus it seems that at least the lowest-order deviations of the
local-disk kinematics from simple expectations based on axisymmetric equilibrium can be attributed entirely to the influence of the Galactic bar.Comment: 10 pages, 8 figures, accepted for publication
in A&
It is investigated if massless particles can couple to scalar fields in a special relativistic theory with classical particles. The only possible obvious theory which is invariant under Lorentz
transformations and reparametrization of the affine parameter leads to trivial trajectories (straight lines) for the massless case, and also the investigation of the massless limit of the massive
theory shows that there is no influence of the scalar field on the limiting trajectories. On the other hand, in contrast to this result, it is shown that massive particles are influenced by the
scalar field in this theory even in the ultra-relativistic limit.Comment: 9 pages, no figures, uses titlepage.sty, LaTeX 2.09 file, submitted to International Journal of Theoretical Physic
The galactic kinematics of Mira variables derived from radial velocities, Hipparcos proper motions and an infrared period-luminosity relation are reviewed. Local Miras in the 145-200day period range
show a large asymmetric drift and a high net outward motion in the Galaxy. Interpretations of this phenomenon are considered and (following Feast and Whitelock 2000) it is suggested that they are
outlying members of the bulge-bar population and indicate that this bar extends beyond the solar circle.Comment: 7 pages, 2 figure, to be published in Mass-Losing Pulsating Stars and their
Circumstellar Matter, Y. Nakada & M. Honma (eds) Kluwer ASSL serie
The Kepler mission has recently discovered a number of exoplanetary systems, such as Kepler-11 and Kepler-32, in which ensembles of several planets are found in very closely packed orbits (often
within a few percent of an AU of one another). These compact configurations present a challenge for traditional planet formation and migration scenarios. We present a dynamical study of the assembly
of these systems, using an N-body method which incorporates a parametrized model of planet migration in a turbulent protoplanetary disc. We explore a wide parameter space, and find that under
suitable conditions it is possible to form compact, close-packed planetary systems via traditional disc-driven migration. We find that simultaneous migration of multiple planets is a viable mechanism
for the assembly of tightly-packed planetary systems, as long as the disc provides significant eccentricity damping and the level of turbulence in the disc is modest. We discuss the implications of
our preferred parameters for the protoplanetary discs in which these systems formed, and comment on the occurrence and significance of mean-motion resonances in our simulations.Comment: 12 pages, 4
figures, 2 tables. Accepted for publication in Monthly Notices of the Royal Astronomical Societ
We present dynamical models based on a study of high-resolution long-slit spectra of the narrow-line region (NLR) in NGC 1068 obtained with the Space Telescope Imaging Spectrograph (STIS) aboard The
Hubble Space Telescope (HST). The dynamical models consider the radiative force due to the active galactic nucleus (AGN), gravitational forces from the supermassive black hole (SMBH), nuclear stellar
cluster, and galactic bulge, and a drag force due to the NLR clouds interacting with a hot ambient medium. The derived velocity profile of the NLR gas is compared to that obtained from our previous
kinematic models of the NLR using a simple biconical geometry for the outflowing NLR clouds. The results show that the acceleration profile due to radiative line driving is too steep to fit the data
and that gravitational forces along cannot slow the clouds down, but with drag forces included, the clouds can slow down to the systemic velocity over the range 100--400 pc, as observed. However, we
are not able to match the gradual acceleration of the NLR clouds from ~0 to ~100 pc, indicating the need for additional dynamical studies.Comment: Paper prepared by emulateapj version 10/09/06 and
accepted for print in Ap
We consider a general method of deprojecting 2D images to reconstruct the 3D structure of the projected object, assuming axial symmetry. The method consists of the application of the Fourier Slice
Theorem to the general case where the axis of symmetry is not necessarily perpendicular to the line of sight, and is based on an extrapolation of the image Fourier transform into the so-called cone
of ignorance. The method is specifically designed for the deprojection of X-ray, Sunyaev-Zeldovich (SZ) and gravitational lensing maps of rich clusters of galaxies. For known values of the Hubble
constant, H0, and inclination angle, the quality of the projection depends on how exact is the extrapolation in the cone of ignorance. In the case where the axis of symmetry is perpendicular to the
line of sight and the image is noise-free, the deprojection is exact. Given an assumed value of H0, the inclination angle can be found by matching the deprojected structure out of two different
images of a given cluster, e.g., SZ and X-ray maps. However, this solution is degenerate with respect to its dependence on the assumed H0, and a third independent image of the given cluster is needed
to determine H0 as well. The application of the deprojection algorithm to upcoming SZ, X-ray and weak lensing projected mass images of clusters will serve to determine the structure of rich clusters,
the value of H0, and place constraints on the physics of the intra-cluster gas and its relation to the total mass distribution.Comment: 7 pages, LaTeX, 2 Postscript figures, uses as2pp4.sty. Accepted
for publication in ApJ Letters. Also available at: http://astro.berkeley.edu:80/~squires/papers/deproj.ps.g
We present 0.55 x 10^6 particle simulations of the accretion of high-density dwarf galaxies by low-density giant galaxies, using models that contain both power-law central density cusps and point
masses representing supermassive black holes. The cusp of the dwarf galaxy is disrupted during the merger, producing a remnant with a central density that is only slightly higher than that of the
giant galaxy initially. Removing the black hole from the giant galaxy allows the dwarf galaxy to remain intact and leads to a remnant with a high central density, contrary to what is observed. Our
results support the hypothesis that the persistence of low-density cores in giant galaxies is a consequence of supermassive black holes.Comment: 5 pages, 2 postscript figures, uses emulateapj.sty.
Accepted for publication in The Astrophysical Journal Letter
We simulate the evolution of one-dimensional gravitating collisionless systems from non- equilibrium initial conditions, similar to the conditions that lead to the formation of dark- matter halos in
three dimensions. As in the case of 3D halo formation we find that initially cold, nearly homogeneous particle distributions collapse to approach a final equilibrium state with a universal density
profile. At small radii, this attractor exhibits a power-law behavior in density, {\rho}(x) \propto |x|^(-{\gamma}_crit), {\gamma}_crit \simeq 0.47, slightly but significantly shallower than the
value {\gamma} = 1/2 suggested previously. This state develops from the initial conditions through a process of phase mixing and violent relaxation. This process preserves the energy ranks of
particles. By warming the initial conditions, we illustrate a cross-over from this power-law final state to a final state containing a homogeneous core. We further show that inhomogeneous but cold
power-law initial conditions, with initial exponent {\gamma}_i > {\gamma}_crit, do not evolve toward the attractor but reach a final state that retains their original power-law behavior in the
interior of the profile, indicating a bifurcation in the final state as a function of the initial exponent. Our results rely on a high-fidelity event-driven simulation technique.Comment: 14 Pages, 13
Figures. Submitted to MNRA
We derive unbiased distance estimates for the Gaia-TGAS data set by correcting for the bias due to the distance dependence of the selection function, which we measure directly from the data. From
these distances and proper motions, we estimate the vertical and azimuthal velocities, W and Vϕ, and angular momentum Lz for stars in the Galactic centre and anticentre directions. The resulting mean
vertical motion W shows a linear increase with both Vϕ and Lz at 10σ significance. Such a trend is expected from and consistent with the known Galactic warp. This signal extends to stars with guiding
centre radii Rg < R0, placing the onset of the warp at R ≲ 7 kpc. At equally high significance, we detect a previously unknown wave-like pattern of W over guiding centre Rg with an amplitude ~1 kms-1
and a wavelength ~2.5 kpc. This pattern is present in both the centre and anticentre directions, consistent with a winding (corrugated) warp or bending wave, likely related to known features in the
outer disc (TriAnd and Monoceros overdensities), and may be caused by the interaction with the Sgr dwarf galaxy ~1Gyr ago. The only significant deviation from this simple fit is a stream-like feature
near Rg ~9 kpc (|Lz| ~2150 kpc km s-1)
We add to the lore of spherical, stellar-system models a two-parameter family with an anisotropic velocity dispersion, and a central point mass (``black hole''). The ratio of the tangential to radial
dispersions, is constant--and constitutes the first parameter--while each decreases with radius as r^{-1/2}. The second parameter is the ratio of the central point mass to the total mass. The Jeans
equation is solved to give the density law in closed form: rho\propto (r/r0)^{-c}/[1+(r/r0)^{3-c}]^2, where r0 is an arbitrary scale factor. The two parameters enter the density law only through
their combination c. At the suggestion of Tremaine, we also explore models with only the root-sum-square of the velocities having a Keplerian run, but with a variable anisotropy ratio. This gives
rise to a more versatile class of models, with analytic expressions for the density law and the dispersion runs, which contain more than one radius-scale parameter.Comment: 10 pages. Final version to
appear in ApJ; minor addition
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Modulo Calculator Overview: Finding Remainders - Powerful and Useful Online Calculators
Modulo Calculator Overview: Finding Remainders
Have you ever wondered how to determine the remainder when you divide one number by another? Understanding how to find remainders is essential in many aspects of mathematics and practical
applications. This is where a modulo calculator comes into play. Let’s take a deep dive into how a modulo calculator works, its foundational concepts, and its applications in the real world.
Modulo Calculator Overview: Finding Remainders
A modulo calculator is a tool designed to find the remainder that results when one number is divided by another. This process is known as modulo operation, often represented as % or mod.
What Can a Modulo Calculator Handle?
A modulo calculator can handle:
• Rational Numbers: Numbers that can be expressed as a fraction.
• Irrational Numbers: Numbers that cannot be expressed as fractions, like π or √2.
• Positive Numbers: Any number greater than zero.
• Negative Numbers: Any number less than zero.
You can calculate these manually or use an online tool for quick results.
Basic Modulo Calculation
The formula used in modulo calculations is straightforward:
[ \text = (\text \times \text) + \text ]
Example: 20 Modulo 3
Consider the expression 20 % 3. According to the formula:
• Let’s find the Quotient first: 20 divided by 3 is 6 with a remainder.
• Multiplying the Quotient by the Divisor: ( 6 \times 3 = 18 )
• Subtract this product from the Dividend: ( 20 – 18 = 2 )
So, 20 % 3 equals 2.
Real-World Examples
Understanding modulo through real-world examples can make the concept much more relatable. Here are two practical scenarios:
Candy Distribution
Imagine you have a bag of candies and you need to divide them among a group of children. Sometimes, you may end up with leftover candies. To find out how many candies will remain, you can use modulo.
Tile Calculation
When tiling bathrooms, you might want to know how many tiles will be left after covering a certain number of bathrooms.
Say you have 62 tiles and need to use 14 tiles per bathroom: [ 62 \div 14 = 4 \text{ R } 6 ]
Thus, ( 62 % 14 = 6 ), leaving you with 6 tiles after tiling the bathrooms.
Manual Calculation Example
For better clarity, let’s break down a manual calculation in detail.
Tiling Case Study
You have 62 tiles and want to cover bathrooms using 14 tiles each.
1. Divide 62 (Dividend) by 14 (Divisor). This results in a quotient of 4.
2. Multiply the quotient by the divisor: ( 4 \times 14 = 56 )
3. Subtract this from the dividend: ( 62 – 56 = 6 )
The remainder is 6, so ( 62 % 14 = 6 ).
Modulo is not limited to theoretical calculations; it finds utility in many everyday tasks and professional fields.
Clock Arithmetic
Modulo is particularly useful in cyclic calculations, like time. For example: [ 251 , \text % 24 = 11 ]
This means 251 hours from now, it will be 11 o’clock.
Even/Odd Numbers
You can determine if a number is even or odd using modulo:
• ( x % 2 = 0 ) (Even number)
• ( x % 2 = 1 ) (Odd number)
Unit Conversion
When dealing with measurement units, modulo is helpful. For instance, converting a larger number of minutes into hours and minutes: [ 135 , \text % 60 = 15 ]
So, 135 minutes is 2 hours and 15 minutes.
Leap Year Calculation
Leap years can be determined using modulo operations.
• If a year is divisible by 4, it’s a leap year (e.g., ( 2020 % 4 = 0 ))
• However, if it’s divisible by 100 (and not 400), it’s not a leap year (e.g., ( 1900 % 100 = 0 ), but ( 1900 % 400 \neq 0 )).
Programming and Modulo
In programming, modulo is frequently used for various tasks:
Common Uses
• Checking Even/Odd: if (x % 2 == 0) { ... }
• Cyclic Tasks: Performing tasks every Nth occurrence.
• Rotation Through Options: Rotating through a list of items.
Example Python Script
Here’s a simple Python script for determining if a year is a leap year:
def is_leap_year(year): if year % 4 == 0: if year % 100 == 0: if year % 400 == 0: return True else: return False else: return True else: return False
Example Usage
year = 2024 print(f” is a leap year: “)
Advanced Uses
Modulo also steps into more complex fields like random number generation and cryptography.
Random Number Generators
Modulo helps in random number generators to ensure the numbers fall within a specific range.
In cryptography, modulo operations are pivotal in creating keys that secure data.
Modulo is a powerful tool for finding remainders in various practical and theoretical contexts. From dividing candies among kids to tiling bathrooms, from programming algorithms to enhancing
cryptographic security, the utility of modulo is extensive. While manual calculations are informative, online tools can simplify even the most complex modulo operations.
For those interested in related tools, consider exploring Basic Calculators, Scientific Calculators, and Integer Calculators to meet all your computational needs.
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Adiabatic expansion of a strongly correlated pure electron plasma
Adiabatic expansion is proposed as a method of increasing the degree of correlation of a magnetically confined pure electron plasma. Quantum mechanical effects and correlation effects make the
physics of the expansion quite different from that for a classical ideal gas. The proposed expansion may be useful in a current experimental effort to cool a pure electron plasma to the liquid and
solid (crystalline) states.
Physical Review Letters
Pub Date:
February 1986
□ Adiabatic Conditions;
□ Crystal Structure;
□ Electron Plasma;
□ Liquid Phases;
□ Plasma Temperature;
□ Quantum Mechanics;
□ Cyclotron Radiation;
□ Entropy;
□ Magnetic Fields;
□ Monte Carlo Method;
□ Vlasov Equations;
□ Plasma Physics;
□ 52.25.Kn;
□ 52.25.Wz;
□ Thermodynamics of plasmas
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Traverse Definitions for Land Surveyors
traverse—A method of surveying in which the lengths and directions of lines between points on the Earth are obtained by or from field measurements and used in determining positions of the points. A
survey traverse may determine the relative positions of the points which it connects in series, and if tied to control stations on an adopted datum, the positions may be referred to that datum.
Survey traverses are classified and identified in a variety of ways: (a) according to methods used, as astronomical traverse; (b) according to quality of results, as first-order traverse; (c)
according to purpose served, as geographical exploration traverse; and (d) according to form, as closed traverse. See also Appendix A, Standards for Geodetic Control Surveys.
traverse, angle-to-left (right)—A technique used in making a survey traverse, wherein all angles are measured in a counterclockwise (clockwise) direction after the surveying instrument has been
oriented by a backsight to the preceding station. The technique is applicable to either closed or open traverses.
traverse, closed—A survey traverse which starts and ends upon the same station, or upon stations whose relative positions have been determined by other surveys of equal or higher order of accuracy.
traverse, open—A survey traverse which begins from a station of known or adopted position but does not end upon such a station; open-end traverse.
traverse, planetable—A graphical traverse executed by a planetable.
traverse, random—A survey traverse run from one survey station to another station which cannot be seen from the first station, in order to determine their relative position.
traverse, stadia—A traverse (transit or planetable) in which distances are measured by the stadia method.
traverse, subtense bar—A traverse method in which course lengths are measured by use of a subtense bar.
traverse, transit—A traverse in which the angles are measured with a transit or theodolite and the lengths with a metal tape. A transit traverse is usually executed for the control of local surveys.
traverse error of closure—See error of closure (definition 5).
traverse line—See line, transit.
traverse station-1 A point in a traverse over which an instrument is placed for measuring (set up). 2 A point which has had its location determined by traverse.
traverse tables—Mathematical tables listing the lengths of the sides opposite the oblique angles for each of a series of right-angle plane triangles as functions of the length and azimuth (or
bearing) of the hypotenuse. Traverse tables are used in computing latitudes and departures in surveying and courses in navigation. One argument of such a table is the angle which the line or course
makes with the meridian (its azimuth or bearing), and the other argument is a distance. In tables used in land surveying, the distance argument is usually a series of integers, from 1 to 9, with
which lines of greater length may be composed. In navigation, the distance argument may run to several hundred miles.
Source: NSPS “Definitions of Surveying and Related Terms“, used with permission.
Part of LearnCST’s exam text bundle.
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Table of Contents
Narrow, gentle, self-avoiding plane-filling trails
On this page we see plane-filling trails visualizing the narrow and gentle, self-avoiding Golden Right Triangle, Golden Isosceles Triangle, Koch Snowflake and Dekking curves.
Golden Right Triangle curve
Definition: (0,√φ,1,-1)(1,0,1,1)(0,-√φ,1,-1)(φ,0,1,1), where φ is the golden ratio ½(1+√5)
This is one of several options to fill a right triangle whose legs have ratio √φ. A colour gradient drawing is also given, showing how fault lines in different sections of the curve line up.
Golden Isosceles Triangle curve
Definition: (1,√(4φ-1),1,-1)(2φ,0,1,1)(-1,-√(4φ-1),1,-1)(2, 0,-1,1), where φ is the golden ratio ½(1+√5).
This is one of several options of filling an isosceles triangle with leg-to-base ratio √φ.
Ventrella's Snowflake Sweep
Definition: Δ(1,0,0,-1,1)(1,0,0,1,-1)(-1,0,1,-1,-1)(0,-1,0,-1,1)(1,0,0,-1,1)(0,0,-1,-1,1)(0,0,-1,1,-1)
Ventrella's “Snowflake Sweep”^, filling the Koch Snowflake. This choice for which sections are filled by a reverse copy of the curve seems to result in the roughest possible fill, with many narrow
passages to deep valleys between higher mountains.
Soft Snowflake Sweep
Definition: Δ(1,0,0,1,1)(1,0,0,-1,-1)(-1,0,1,1,-1)(0,-1,0,1,1)(1,0,0,1,1)(0,0,-1,1,1)(0,0,-1,-1,-1)
In the definition of this curve, all sections are reversed as compared to the definition of the previous curve—resulting in a “softer” terrain with fewer narrow passages to deep valleys.
Dekking's curve
Definition: (1,0,1,1)(1,0,1,1)(0,1,-1,-1)(-1,0,-1,-1)(0,1,1,1)(1,0,1,1)(0,1,-1,-1)(-1,0,-1,-1)(-1,0,1,1)(0,1,-1,-1)(1,0,1,1)(1,0,1,1)(1,0,-1,-1)(0,1,1,1)(1,0,-1,-1)(0,-1,1,1)(0,-1,1,1)(-1,0,-1,-1)
A curve by Dekking^ that has Arrwwid number^ 3, which is unusual for curves based on a subdivision into squares.
^2) F. M. Dekking. Recurrent sets. Advances in Mathematics 44, pages 78–104, 1982.
^3) H. Haverkort: Recursive tilings and space-filling curves with little fragmentation. Journal of Computational Geometry 2(1), page 92-127, 2011.
pftrails/narrow_gentle_curves.txt · Last modified: 2020/02/15 00:25 by administrator
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[l2h] problem with bibitem in thebibliography
Peter Morling pmorling at nat.sdu.dk
Mon Feb 2 14:53:37 CET 2004
i have a problem, the following (tex):
\bibitem{} Godambe, P.V. (1960). An optimum property of regular maximum
likelihood estimation. \emph{Ann. Math. Statist.} \textbf{81}, 1208--1212.
[Shows optimality of one-parameter score function.]
\bibitem{} Durbin (1960).
\bibitem{} Bhapkar, V.P. (1972). On a measure of efficiency in an estimating
equation. \emph{Sankhy\={a}} A \textbf{34}, 467--472.
will create the correct numbering in latex [1], [2] and [3] and you can use
cite to referer to each of them. But, after running L2H, each will be
numbered with [3]. What is the problem?
Programmer Peter Morling, University of Southern Denmark
Department of Statistics, Sdr. Boulevard 23A, DK-5000 Odense C
Phone (+45) 6550 3399
More information about the latex2html mailing list
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McGraw Hill Math Grade 8 Lesson 3.1 Answer Key Changing Improper Fractions to Mixed Numbers
Practice the questions of McGraw Hill Math Grade 8 Answer Key PDF Lesson 3.1 Changing Improper Fractions to Mixed Numbers to secure good marks & knowledge in the exams.
McGraw-Hill Math Grade 8 Answer Key Lesson 3.1 Changing Improper Fractions to Mixed Numbers
Exercises Convert to a Mixed Number
Question 1.
Given to convert \(\frac{64}{3}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{21 X 3 + 1}{3}\),
therefore we get 21\(\frac{1}{3}\).
Question 2.
Given to convert \(\frac{101}{4}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{25 X 4 + 1}{4}\),
therefore we get 25\(\frac{1}{4}\).
Question 3.
Given to convert \(\frac{15}{2}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{7 X 2 + 1}{2}\),
therefore we get 7\(\frac{1}{2}\).
Question 4.
Given to convert \(\frac{52}{3}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{17 X 3 + 1}{3}\),
therefore we get 17\(\frac{1}{3}\).
Question 5.
Given to convert 5\(\frac{6}{12}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{5 X 12 + 6}{12}\),
therefore we get 5\(\frac{6}{12}\).
Question 6.
Given to convert \(\frac{137}{11}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{12 X 11 + 5}{11}\),
therefore we get 12\(\frac{5}{11}\).
Question 7.
Given to convert \(\frac{176}{16}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{11 X 16}{16}\) = 11.
Question 8.
Given to convert \(\frac{61}{8}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{7 X 8 + 5}{8}\),
therefore we get 7\(\frac{5}{8}\).
Question 9.
Given to convert \(\frac{121}{21}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{5 X 21 + 16}{21}\),
therefore we get 5\(\frac{16}{21}\).
Question 10.
Given to convert \(\frac{53}{2}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{26 X 2 + 1}{2}\),
therefore we get 26\(\frac{1}{2}\).
Question 11.
Given to convert \(\frac{49}{11}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{4 X 11 + 5}{11}\),
therefore we get 4\(\frac{5}{11}\).
Question 12.
Given to convert \(\frac{312}{19}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{16 X 19 + 8}{19}\),
therefore we get 16\(\frac{8}{19}\).
Question 13.
Given to convert \(\frac{98}{8}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{12 X 8 + 2}{8}\),
therefore we get 12\(\frac{2}{8}\).
Question 14.
Given to convert \(\frac{87}{7}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{12 X 7 + 3}{7}\),
therefore we get 12\(\frac{3}{7}\).
Question 15.
Given to convert \(\frac{159}{12}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{13 X 12 + 3}{12}\),
therefore we get 13\(\frac{3}{12}\).
Question 16.
Given to convert \(\frac{360}{16}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{22 X 16 + 8}{16}\),
therefore we get 22\(\frac{8}{16}\).
Question 17.
Given to convert \(\frac{74}{3}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{24 X 3 + 2}{3}\),
therefore we get 24\(\frac{2}{3}\).
Question 18.
Given to convert \(\frac{71}{4}\) to a mixed number,
As numerator is greater than denominator so we write in
mixed fraction as \(\frac{17 X 4 + 3}{4}\),
therefore we get 17\(\frac{3}{4}\).
Question 19.
Gerrie collects honey from a few beehives. She scoops out the honey with a small jar that holds \(\frac{1}{3}\) of a cup.
Over the last two weeks Gerrie has filled this jar 158 times. How many cups of honey has she collected?
52\(\frac{2}{3}\) cups of honey,
Given Gerrie collects honey from a few beehives.
She scoops out the honey with a small jar that holds \(\frac{1}{3}\) of a cup. Over the last two weeks Gerrie has filled this jar 158 times.
So many cups of honey has she collected are
158 X \(\frac{1}{3}\) = \(\frac{158}{3}\)
numerator is greater than denominator so we write in
mixed fraction as \(\frac{52 X 3 + 2}{3}\),
therefore we get 52\(\frac{2}{3}\).
Question 20.
To finish sewing her tapestry, Petra needs 142 strips of cloth that are each one quarter of a yard. How many yards of cloth is that?
35\(\frac{2}{4}\) yards of cloth,
Given to finish sewing her tapestry, Petra needs 142 strips of
cloth that are each one quarter of a yard.
So many yards of cloth is that 142 X \(\frac{1}{4}\) = \(\frac{142}{4}\) numerator is greater than denominator,
so we write in mixed fraction as \(\frac{35 X 4 + 2}{4}\),
therefore we get 35\(\frac{2}{4}\).
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Highlight dates that are weekends in Excel November 6, 2024 - Excel Office
Highlight dates that are weekends in Excel
This tutorial shows how to Highlight dates that are weekends in Excel using the example below;
If you want to use conditional formatting to highlight dates occur on weekends (i.e. Saturday or Sunday), you can use a simple formula based on the WEEKDAY function.
For example, if you have dates in the range C4:C10, and want to weekend dates, select the range C4:C10 and create a new conditional formatting rule that uses this formula:
Note: it’s important that CF formulas be entered relative to the “active cell” in the selection, which is assumed to be C5 in this case.
Once you save the rule, you’ll see all dates that are a Saturday or a Sunday highlighted by your rule.
How this formula works
This formula uses the WEEKDAY function to test dates for either a Saturday or Sunday. When given a date, WEEKDAY returns a number 1-7, for each day of the week. In it’s standard configuration,
Saturday = 7 and Sunday = 1. By using the OR function, use WEEKDAY to test for either 1 or 7. If either is true, the formula will return TRUE and trigger the conditional formatting.
Highlighting the entire row
If you want to highlight the entire row, apply the conditional formatting rule to all columns in the table and lock the date column:
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Solving the division table of 9
The students learn to solve the division table of 9.
Students will be able to solve the division table of 9.
You ask student volunteers to come up to the whiteboard and take jumps of 9 with the monkey. You do this forwards and backwards. After this you ask the students to draw a pizza and divide this into 9
even pieces. Next have the students ask a friend which piece of pizza they would choose and why.
First you discuss the importance of solving division problems. Next you explain that with a division problem you have a certain amount of something in each group and that in the division table of 9
you are going to divide by 9 each time. For this you show an example of 18 pieces of pizza, that you are going to divide evenly into 9 slices. From here you can use the bar on the bottom right to
navigate through the lesson (visual, abstract, story) in a way that best suits your learning goals. Otherwise you continue following the pages of the lesson in order and move on to problems with
visual support. Here you explain how you calculate a division problem by telling how a division problem words with images of chewing gum that must be divided between 9 mouths, then you practice
dividing a number of candles between 9 cakes. After this you show the images and the students must come up with the problem that matches each one. Next you have the students solve division problems
on their own with help from an image. After this you discuss the abstract problems. First you show what division by 9 looks like when you divide the number 9 evenly. After this you show the problems
from the division table of 9 and show that you can use the division table of 9 to solve them. Next the students practice using what they have learned. Then you walk the students through the steps of
solving a story problem. The students can click on the speaker to hear the story.Check whether the students can solve problems from the division table of 9 by asking them the following questions:-
What does the problem 18 ÷ 9 mean? Can you name an example?- What can be a memory aid when solving the division table of 9?
The students test their understanding of the division table of 9 through ten exercises. For some of the exercises there is visual support, others are abstract division problems and other exercises
are story problems about the division table of 9.
You have the students solve one more problem with visual support, then another abstract problem and then they answer another story problem. To close you can play a ball game. A student names a
problem from the division table of 9 and throws the ball to another student. The person who caught the ball must answer the problem, name another problem from the division table of 9, and then pass
the ball along to a different student.
You can have the students use concrete materials to help them solve division problems, by dividing them up into 9 groups.
MAB blocks or other concrete materials for each pair.A ball.
Gynzy is an online teaching platform for interactive whiteboards and displays in schools.
With a focus on elementary education, Gynzy’s Whiteboard, digital tools, and activities make it easy for teachers to save time building lessons, increase student engagement, and make classroom
management more efficient.
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12-17-11 - LZ Optimal Parse with A Star Part 2
Okay, optimal parsing with A star. (BTW "optimal" parsing here is really a misnomer that goes back to the LZSS backwards parse where it really was optimal; with a non-trivial coder you can't really
do an optimal parse, we really mean "more optimal" (than greedy/lazy type parses)).
Part 1 was just a warmup, but may get you in the mood.
The reason for using A Star is to handle LZ parsing when you have adaptive state. The state changes as you step through the parse forward, so it's hard to deal with this in an LZSS style backwards
parse. See some previous notes on backwards parsing and LZ here : 1 , 2 , 3
So, the "state" of the coder is something like maybe an adaptive statistical mode, maybe the LZMA "markov chain" state machine variable, maybe an LZX style recent offset cache (also used in LZMA). I
will assume that the state can be packed into a not too huge size, maybe 32 bytes or so, but that the count of states is too large to just try them all (eg. more than 256 states). (*1)
(*1 - in the case that you can collapse the entire state of the coder into a reasonably small number of states (256 or so) then different approaches can be used; perhaps more on this some day; but
basically any adaptive statistical state or recent offset makes the state space too large for this).
Trying all parses is impossible even for the tiniest of files. At each position you have something like 1-16 options. (actually sometimes more than 16, but you can limit the choices without much
penalty (*2)). You always have the choice of a literal, when you have a match there are typically several offsets, and several lengths per offset to consider. If the state of the coder is changed by
the parse choice, then you have to consider different offsets even if they code to the same number of bits in the current decision, because they affect the state in the future.
(*2 - the details of this depend on the back end of coder; for example if your offset coder is very simple, something like just Golomb type (NOSB) coding, then you know that only the shortest offset
for a given length needs to be considered, another simplification used in LZMA, only the longest length for a given offset is considered; in some coders it helps to consider shorter length choices as
well; in general for a match of Length L you need to consider all lengths in [2,L] but in practice you can reduce that large set by picking a few "inflection points" (perhaps more on this some day)).
Okay, a few more generalities. Let's revisit the LZSS backwards optimal parser. It came from a forward style parser, which we can implement with "dynamic programming" ; like this :
At pos P , consider the set of possible coding choices {C}
For each choice (ci), find the cost of the choice, plus the cost after that choice :
Cost to end [ci] = Current cost of choice C [ci] + Best cost to end [ P + C[ci].len ]
choose ci as best Cost to end
Best code to end[ P ] = Cost to end [ best ci ]
You may note that if you do this walking forward, then the "Best cost to end" at the next position may not be computed yet. If so, then you suspend the current computation and step ahead to do that,
then eventually come back and finish the current decision.
Of course with LZSS the simpler way to do it is just to parse backwards from the end, because that ensures the future costs are already done when you need them. But let's stick with the forward parse
because we need to introduce adaptive state.
The forward parse LZSS (with no state) is still O(N) just like the backward parse (this time cost assumes the string matching is free or previously done, and that you consider a fixed number of match
choices, not proportional to the number of matches or length of matches, which would ruin the O(N) property) - it just requires more book keeping.
In full detail a forward LZSS looks like this :
Set "best cost to end" for all positions to "uncomputed"
Push Pos 1 on stack of needed positions.
While stack is not empty :
pop stack; gives you a pos P
If any of the positions that I need ( P + C.len ) are not done :
push self (P) back on stack
push all positions ( P + C.len ) on stack
in order from lowest to highest pos
make a choice as above and fill "best cost to end" at pos P
If you could not make a choice the first time you visit pos P, then because of the order that we push things on the stack, when you come back and pop P the second time it's gauranteed that everything
needed is done. Therefore each position is visited at most twice. Therefore it's still O(N).
We push from lowest to highest len, so that the pops are highest pos first. This makes us do later positions first; that way earlier positions are more likely to have everything they need already
Of course with LZSS this is silly, you should just go backwards, but we'll use it to inspire the next step.
To be continued...
No comments:
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This paper uses elementary methods to derive the formulas for and to tablulate (in the case q = 2) two related q-analogs of the Stirling numbers of the second kind and the Bell numbers for direct-sum
decompositions (vector space analogs of set partitions) of a finite vector space over a finite field with q elements.
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3D Angular Menu
Three Points Angle
Measures an angle defined by two segments with a common point. The second point you select will be the vertex of the angle.
Once the markup label is displayed, you can also determine the reflex angle by using the Flip Axis command on the Markup context menu.
See Working with Angular Measurements for information on using this command and the context menu.
The Flip Axis command works differently on a three points angle and a two points draft angle.
• On a three points angle, the Flip Axis command measures the angle in the opposite direction.
• On a two points draft angle, the Flip Axis command flips the direction of the X-, Y-, or Z-axis segment used to form the angle.
Two Edge Angle
The Two Edge Angle command measures an angle defined by two edges on the model. The two edges picked must be co-planar.
See Working with Angular Measurements for information on using this command and the context menu.
Two Surfaces Angle
The Two Surfaces Angle command measures an angle defined by two surfaces on the model. The two surfaces picked must be
• both planar non-parallel surfaces,
• one cylindrical and one planar surface.
See Working with Angular Measurements for information on using this command and the context menu.
Surface Point to Point Angle
The Surface Point to Point Angle command measures an angle defined by the line between a point and a surface point and the normal vector of the surface. Every surface has a normal, that is, a vector
perpendicular to the surface.
See Working with Angular Measurements for information on using this command and the context menu.
Surface Point to Edge Angle
The Surface Point to Edge Angle command measures an angle defined by a point on the surface and a straight edge on the model.
See Working with Angular Measurements for information on using this command and the context menu.
Surface Point to Surface Point Angle
The Surface Point to Surface Point Angle command measures the angle defined by points on two surfaces.
See Working with Angular Measurements for information on using this command and the context menu.
Surface Point to Circle Tangent Angle
The Surface Point to Circle Tangent Angle command measures the angle defined by a point on a surface and the tangent of a selected circle.
See Working with Angular Measurements for information on using this command and the context menu.
Two Points Draft Angle
The Two Points Draft Angle command measures an angle defined by a line between two selected points and the X-, Y-, or Z-axis.
Once the markup label is displayed, you can use the Flip Axis command on the Markup context menu to change the direction of the X-, Y-, or Z-axis segment used to form the angle. Right-click the label
and select Flip Axis from the context menu.
See Working with Angular Measurements for information on using this command and the context menu.
The Flip Axis command works differently on a three points angle and a two points draft angle.
• On a three points angle, the Flip Axis command measures the angle in the opposite direction.
• On a two points draft angle, the Flip Axis command flips the direction of the X-, Y-, or Z-axis segment used to form the angle.
Surface Draft Angle
The Surface Draft Angle command measures an angle defined by a surface point and the X-, Y-, or Z-axis.
See Working with Angular Measurements for information on using this command and the context menu.
Related Topics
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How do you find the derivative of log _ 3 (2x) / x^2? | HIX Tutor
How do you find the derivative of #log _ 3 (2x) / x^2#?
Answer 1
$f ' \left(x\right) = \frac{1 - 2 \ln \left(2 x\right)}{{x}^{3} \ln \left(3\right)}$
First, rewrite the logarithm in terms of the natural logarithm, since we know how to differentiate the natural logarithm.
It can be rewritten using the change of base formula: #log_3(2x)=ln(2x)/ln(3)#
Thus, the function can be written as
To differentiate this, use the quotient rule.
We can find each of these derivatives individually:
The first requires the chain rule:
This can also be found through first splitting up #ln(2x)=ln(2)+ln(x)#...
To find the second, just use the power rule and remember that #ln(3)# is just a constant.
Plugging both these back in yields
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
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% Encoding: UTF-8 @COMMENT{BibTeX export based on data in FAU CRIS: https://cris.fau.de/} @COMMENT{For any questions please write to cris-support@fau.de} @incollection{faucris.222693177, abstract =
{We consider the out-of-the-plane displacements of nonlinear elastic strings which are coupled through point masses attached to the ends and viscoelastic springs. We provide the modeling, the
well-posedness in the sense of classical semi-global C^2 -solutions together with some extra regularity at the masses and then prove exact boundary controllability and velocity-feedback
stabilizability, where controls act on both sides of the mass-spring-coupling.}, author = {Leugering, Günter and Li, Tatsien and Wang, Yue}, booktitle = {Springer INdAM Series}, doi = {10.1007/
978-3-030-17949-6{\_}8}, faupublication = {yes}, keywords = {Coupled system of quasilinear wave equations; Dynamical boundary condition; Exact boundary controllability; Visoelastic springs}, note =
{CRIS-Team Scopus Importer:2019-07-19}, pages = {139-156}, peerreviewed = {unknown}, publisher = {Springer International Publishing}, series = {Springer INdAM Series}, title = {1-d {Wave} {Equations}
{Coupled} via {Viscoelastic} {Springs} and {Masses}: {Boundary} {Controllability} of a {Quasilinear} and {Exponential} {Stabilizability} of a {Linear} {Model}}, volume = {32}, year = {2019} }
@article{faucris.110564784, abstract = {In this paper we consider a special optimization problem with two objectives which arises in antenna theory. It is shown that this abstract bicriterial
optimization problem has at least one solution. Discretized versions of this problem are also discussed, and the relationships between these finite dimensional problems and the infinite dimensional
problem are investigated. Moreover, we present numerical results for special parameters using a multiobjective optimization method.}, author = {Jahn, Johannes and Jüschke, A. and Kirsch, A.},
faupublication = {yes}, journal = {Computational Optimization and Applications}, keywords = {Antenna theory; Multiobjective optimization}, note =
{UnivIS-Import:2015-03-05:Pub.1997.nat.dma.pama21.abicri}, pages = {261-276}, peerreviewed = {Yes}, title = {{A} {Bicriterial} {Optimization} {Problem} of {Antenna} {Design}}, volume = {7}, year =
{1997} } @article{faucris.118132344, abstract = {In this paper properly minimal elements of a set are characterized as minimal solutions of appropriate approximation problems without any convexity
assumptions.}, author = {Jahn, Johannes}, faupublication = {no}, journal = {SIAM Journal on Control and Optimization}, note = {UnivIS-Import:2015-03-05:Pub.1985.nat.dma.pama21.achara}, pages =
{649-656}, peerreviewed = {Yes}, title = {{A} {Characterization} of {Properly} {Minimal} {Elements} of a {Set}}, volume = {23}, year = {1985} } @article{faucris.123273084, abstract = {We investigate
the propagation of cracks in 2-d elastic domains, which are subjected to quasi-static loading scenarios. As we take cohesive effects along the crack path into account and impose a non-penetration
condition, inequalities appear in the constitutive equations describing the elastic behavior of a domain with crack. In contrast to existing approaches, we consider cohesive effects arising from
crack opening in normal as well as in tangential direction. We establish a constrained energy minimization problem and show that the solution of this problem satisfies the set of constitutive
equations. In order to solve the energy minimization problem numerically, we apply a finite element discretization using a combination of standard continuous finite elements with so-called cohesive
elements. A particular strength of our method is that the crack path is a result of the minimization process. We conclude the article by numerical experiments and compare our results to results given
in the literature.}, author = {Prechtel, Marina and Leugering, Günter and Steinmann, Paul and Stingl, Michael}, doi = {10.3934/cpaa.2013.12.1705}, faupublication = {yes}, journal = {Communications on
Pure and Applied Analysis}, keywords = {Energy minimization; cohesive elements; crack problems.}, pages = {1705-1729}, peerreviewed = {Yes}, title = {{A} cohesive crack propagation model:
{Mathematical} theory and numerical solution}, volume = {12}, year = {2013} } @article{faucris.122281104, abstract = {We propose a novel approach to adaptive refinement in FEM based on local
sensitivities for node insertion. To this end, we consider refinement as a continuous graph operation, for instance by splitting nodes along edges. Thereby, we introduce the concept of the
topological mesh derivative for a given objective function. For its calculation, we rely on the first-order asymptotic expansion of the Galerkin solution of a symmetric linear second-order elliptic
PDE. In this work, we apply this concept to the total potential energy, which is related to the approximation error in the energy norm. In fact, our approach yields local sensitivities for
minimization of the energy error by refinement. Moreover, we prove that our indicator is equivalent to the classical explicit a posteriori error estimator in a certain sense. Numerical results
suggest that our method leads to efficient and competitive adaptive refinement.}, author = {Leugering, Günter and Friederich, Jan and Steinmann, Paul}, faupublication = {yes}, journal = {Control and
Cybernetics}, keywords = {Adaptive mesh refinement; Asymptotic expansion; Sensitivity-based refinement; Topological sensitivity}, pages = {279-306}, peerreviewed = {Yes}, title = {{Adaptive} finite
elements based on sensitivities for topological mesh changes}, url = {https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85018838294&origin=inward}, volume = {43}, year = {2014} }
@article{faucris.122284184, abstract = {We consider refinement of finite element discretizations by splitting nodes along edges. For this process, we derive asymptotic expansions of Galerkin
solutions of linear second-order elliptic equations. Thereby, we calculate a topological derivative w.r.t. node insertion for functionals such as the total potential energy, minimization of which
decreases the approximation error in the energy norm. Hence, these sensitivities can be used to define indicators for local h-refinement. Our results suggest that this procedure leads to an efficient
adaptive refinement method. This presentation is concerned with a model problem in 1d. The extension of this concept to higher dimensions will be the subject of forthcoming publications. © 2012
WILEY-VCH Verlag GmbH & Co. KGaA.}, author = {Leugering, Günter and Friederich, Jan and Steinmann, Paul}, doi = {10.1002/gamm.201210012}, faupublication = {yes}, journal = {GAMM-Mitteilungen}, pages
= {175-190}, peerreviewed = {unknown}, title = {{Adaptive} refinement based on asymptotic expansions of finite element solutions for node insertion in 1d}, volume = {35}, year = {2012} } @article
{faucris.117895184, abstract = {We show that the generator of the semi-group associated with an integro-partial differential equation can be decomposed by applying a similarity transform. This
transformation accomplishes a spectral decomposition and has, therefore, various important implications from a control-theoretic point of view. Examples are given, including certain singular and
regular-type viscoelastic materials as well as thermoelastic models of beams.}, author = {Leugering, Günter}, faupublication = {no}, journal = {Journal De Mathematiques Pures Et Appliquees}, keywords
= {Integrodifferential equation; Partial differential equation; Hilbert space; Operator equation; Controllability; Stabilizability; Viscoelastic plate; Thermoelasticity; Spectral decomposition},
month = {Jan}, pages = {561-587}, peerreviewed = {Yes}, title = {{A} {DECOMPOSITION} {METHOD} {FOR} {INTEGRO}-{PARTIAL} {DIFFERENTIAL}-{EQUATIONS} {AND} {APPLICATIONS}}, volume = {71}, year = {1992}
} @article{faucris.109450484, author = {Jahn, Johannes}, doi = {10.1007/s10589-014-9674-8}, faupublication = {yes}, journal = {Computational Optimization and Applications}, note =
{UnivIS-Import:2015-03-09:Pub.2014.nat.dma.pama21.aderiv}, pages = {393 - 411}, peerreviewed = {Yes}, title = {{A} derivative-free descent method in set optimization}, volume = {60}, year = {2015} }
@article{faucris.122286384, author = {Leugering, Günter and Pflug, Lukas and Spinola, Michele}, doi = {10.1002/cite.201650478}, faupublication = {yes}, journal = {Chemie Ingenieur Technik}, pages =
{1366-1367}, peerreviewed = {unknown}, title = {{Adjungierten}-basierte {Prozessoptimierung} der {Nanopartikel}-{Synthese}}, volume = {88}, year = {2016} } @inproceedings{faucris.223719326, author =
{Engel, Ulf and Meßner, Arthur and Geiger, Manfred}, booktitle = {Advanced Technology of Plasticity 1996, Proc. Of the 5th ICTP}, editor = {ALTAN, T.}, faupublication = {yes}, note = {LFT
Import::2019-08-05 (587)}, pages = {903-907}, peerreviewed = {unknown}, title = {{Advanced} {Concept} for the {FE}-{Simulation} of {Metal} {Forming} {Processes} for the {Production} of
{Microparts}.}, year = {1996} } @article{faucris.122287044, abstract = {In this paper we present a chain of mathematical models that enables the numerical simulation of the airlay process and the
investigation of the resulting nonwoven material by means of virtual tensile strength tests. The models range from a highly turbulent dilute fiber suspension flow to stochastic surrogates for fiber
lay-down and web formation and further to Cosserat networks with effective material laws. Crucial is the consistent mathematical mapping between the parameters of the process and the material. We
illustrate the applicability of the model chain for an industrial scenario, regarding data from computer tomography and experiments. By this proof of concept we show the feasibility of future
simulation-based process design and material optimization which are long-term objectives in the technical textile industry.}, author = {Gramsch, Simone and Klar, Axel and Leugering, Günter and
Marheineke, Nicole and Nessler, Christian and Strohmeyer, Christoph and Wegener, Raimund}, doi = {10.1186/s13362-016-0034-4}, faupublication = {yes}, journal = {Journal of Mathematics in Industry},
keywords = {airlay process; effective material laws; fiber lay-down; fiber networks; fiber suspension flow; homogenization; model chain; nonwoven material; stochastic surrogates; virtual tensile
strength test}, peerreviewed = {unknown}, title = {{Aerodynamic} web forming: process simulation and material properties}, volume = {6}, year = {2016} } @article{faucris.230759798, abstract = {Water
wave propagation in an open channel network can be described by the viscous Burgers' equation on the corresponding connected graph, possibly with small viscosity. In this paper, we propose a fast
adaptive spectral graph wavelet method for the numerical solution of the viscous Burgers' equation on a star-shaped connected graph. The vital feature of spectral graph wavelets is that they can be
constructed on any complex network using the graph Laplacian. The essence of the method is that the same operator can be used for the construction of the spectral graph wavelet and the approximation
of the differential operator involved in the Burgers' equation. In this paper, two test problems are considered with homogeneous Dirichlet boundary condition. The numerical results show that the
method accurately captures the evolution of the localized patterns at all the scales, and the adaptive node arrangement is accordingly obtained. The convergence of the given method is verified, and
efficiency is shown using CPU time.}, author = {Shukla, Ankita and Mehra, Mani and Leugering, Günter}, doi = {10.1002/mma.5907}, faupublication = {yes}, journal = {Mathematical Methods in the Applied
Sciences}, note = {CRIS-Team WoS Importer:2019-12-20}, peerreviewed = {Yes}, title = {{A} fast adaptive spectral graph wavelet method for the viscous {Burgers}' equation on a star-shaped connected
graph}, year = {2019} } @article{faucris.110509124, abstract = {In many decision problems, criteria occur that can be expressed as ratios. The corresponding optimization problems are nonconvex
programs of fractional type. In this paper, an algorithm for the numerical solution of these problems is introduced that converges always at superlinear speed. Numerical examples are presented.},
author = {Gugat, Martin}, doi = {10.1287/mnsc.42.10.1493}, faupublication = {no}, journal = {Management Science}, keywords = {Algorithm; Convex Auxiliary Problem; Decision Analysis; Dual Solutions;
Generalized Fractional Programs; Multiple Criteria; Superlinear Convergence}, note = {UnivIS-Import:2015-03-05:Pub.1996.nat.dma.lama1.afasta}, pages = {1493-1499}, peerreviewed = {Yes}, title = {{A}
{Fast} {Algorithm} for a {Class} of {Generalized} {Fractional} {Programs}}, volume = {42}, year = {1996} } @article{faucris.115132644, abstract = {In 1953, K.J. Arrow, E.W. Barankin, and D. Blackwell
proved a famous theorem concerning the density of the set of minimal solutions of strictly positive support functionals in the set of minimal elements of a compact convex subset of R^n. This result
has some important and interesting consequences in multi-objective optimization. But this theorem is restricted to the space R^n partially ordered with respect to the componentwise ordering. In this
paper, it is shown that the Arrow-Barankin-Blackwell theorem remains true in a real normed space partially ordered by a Bishop-Phelps cone.}, author = {Jahn, Johannes}, faupublication = {yes},
journal = {SIAM Journal on Control and Optimization}, note = {UnivIS-Import:2015-03-05:Pub.1988.nat.dma.pama21.agener}, pages = {999-1005}, peerreviewed = {Yes}, title = {{A} {Generalization} of a
{Theorem} of {Arrow}, {Barankin}, and {Blackwell}}, volume = {26}, year = {1988} } @article{faucris.121539044, author = {Jahn, Johannes}, faupublication = {no}, journal = {Optimization}, note =
{UnivIS-Import:2015-03-05:Pub.1981.nat.dma.pama21.agloba}, pages = {493-511}, peerreviewed = {Yes}, title = {{A} {Globally} {Convergent} {Method} for {Nonlinear} {Programming}}, volume = {12}, year =
{1981} } @inproceedings{faucris.123511344, abstract = {A highly efficient decoding algorithm for the REMOS (REverberationMOdeling for Speech recognition) concept for distant-talking speech
recognition as proposed in [1] is suggested to reduce the computational complexity by about two orders of magnitude and thereby allowing for first real-time implementations. REMOS is based on a
combined acoustic model consisting of a conventional hidden Markov model (HMM), modeling the clean speech, and a reverberation model. During recognition, the most likely clean-speech and reverberant
contributions are estimated by solving an inner optimization problem for logarithmic melspectral (logmelspec) features. In this paper, two approximation techniques for the inner optimization problem
are derived. Connected digit recognition experiments confirm that the computational complexity is significantly reduced. Ensuring that the global optima of the inner optimization problem are found,
the decoding algorithm based on the proposed approximations even increases the recognition accuracy relative to interior point optimization techniques. © EURASIP, 201}, author = {Maas, Roland and
Sehr, Armin and Gugat, Martin and Kellermann, Walter}, booktitle = {18th European Signal Processing Conference, EUSIPCO 2010}, date = {2010-08-23/2010-08-27}, doi = {10.5281/zenodo.41948},
faupublication = {yes}, pages = {1983-1987}, peerreviewed = {Yes}, title = {{A} highly efficient optimization scheme for remos-based distant-talking speech recognition}, venue = {Aalborg}, year =
{2010} } @incollection{faucris.118333644, address = {Berlin}, author = {Jahn, Johannes}, booktitle = {Operations Research Proceedings 1987}, editor = {H. Schellhaas, P. van Beek, H. Isermann, R.
Schmidt, M. Zijlstra}, faupublication = {yes}, note = {UnivIS-Import:2015-04-17:Pub.1988.nat.dma.pama21.ametho}, pages = {576-587}, peerreviewed = {unknown}, publisher = {Springer}, title = {{A}
{Method} of {Reference} {Point} {Approximation} in {Vector} {Optimization}}, year = {1988} } @article{faucris.113416204, abstract = {
We consider refinement of finite element discretizations by splitting nodes along edges. For
this process, we derive asymptotic expansions of Galerkin solutions of linear second-order
elliptic equations. Thereby, we calculate a topological derivative w.r.t. node insertion for
functionals such as the total potential energy, minimization of which decreases the approxi-
mation error in the energy norm. Hence, these sensitivities can be used to define indicators for
local h-refinement. Our results suggest that this procedure leads to an efficient adaptive re-
finement method. This presentation is concerned with a model problem in 1d. The extension
of this concept to higher dimensions will be the subject of forthcoming publications.
}, author = {Friederich, Jan and Leugering, Günter and Steinmann, Paul}, doi = {10.1002/gamm.201210012}, faupublication = {yes}, journal = {GAMM-Mitteilungen}, note =
{UnivIS-Import:2015-03-09:Pub.2013.tech.FT.FT-TM.anadap}, pages = {175-190}, peerreviewed = {unknown}, title = {{An} adaptive finite element method based on sensitivities for node insertion}, url =
{http://onlinelibrary.wiley.com/doi/10.1002/gamm.201210012/full}, volume = {35}, year = {2012} } @article{faucris.249332696, abstract = {In this article, we propose an adaptive spectral graph wavelet
method to solve partial differential equations on network-like structures using so-called spectral graph wavelets. The concept of spectral graph wavelets is based on the discrete graph Laplacian. The
beauty of the method lies in the fact that the same operator is used for the approximation of differential operators and for the construction of the spectral graph wavelets. Two test functions on
different topologies of the network are considered in order to explain the features of the spectral graph wavelet (i.e., behavior of wavelet coefficients and reconstruction error). Subsequently, the
method is applied to parabolic problems on networks with different topologies. The numerical results show that the method accurately captures the emergence of the localized patterns at all the scales
(including the junction of the network) and the node arrangement is accordingly adapted. The convergence of the method is verified and the efficiency of the method is discussed in terms of CPU
time.}, author = {Mehra, Mani and Shukla, Ankita and Leugering, Günter}, doi = {10.1007/s10444-020-09824-9}, faupublication = {yes}, journal = {Advances in Computational Mathematics}, keywords =
{Adaptive node arrangement; Network models; Spectral graph wavelet}, note = {CRIS-Team Scopus Importer:2021-02-12}, peerreviewed = {Yes}, title = {{An} adaptive spectral graph wavelet method for
{PDEs} on networks}, volume = {47}, year = {2021} } @article{faucris.110513084, abstract = {The problem of rational approximation is facilitated by introducing both lower and upper bounds on the
denominators. For a general fractional inf-sup problem with constrained denominators, a differential correction algorithm and convergence results are given. Numerical examples are presented. The
proposed algorithm has certain advantages compared with the original differential correction method: not only upper but also lower bounds for the optimal value are computed, linear convergence is
always guaranteed, and due to a different start convergence is more rapid.}, author = {Gugat, Martin}, doi = {10.1007/s003659900010}, faupublication = {no}, journal = {Constructive Approximation},
keywords = {41A20; 65D15; 90C32; 90C34; Chebyshev approximation; Constrained denominators; Differential correction algorithm; Fractional programming; Rational approximation}, note =
{UnivIS-Import:2015-03-05:Pub.1996.nat.dma.lama1.analgo}, pages = {197-221}, peerreviewed = {Yes}, title = {{An} {Algorithm} for {Chebyshev} {Approximation} by {Rationals} with {Constrained}
{Denominators}}, volume = {12}, year = {1996} } @article{faucris.124194444, abstract = {We consider a system of scalar nonlocal conservation laws on networks that model a highly re-entrant
multi-commodity manufacturing system as encountered in semi-conductor production. Every single commodity is modeled by a nonlocal conservation law, and the corresponding PDEs are coupled via a
collective load, the work m progress. We illustrate the dynamics for two commodities. In the applications, directed acyclic networks naturally occur, therefore this type of networks is considered. On
every edge of the network we have a system of coupled conservation laws with nonlocal velocity. At the junctions the right hand side boundary data of the foregoing edges is passed as left hand side
boundary data to the following edges and PDEs. For distributing junctions, where we have more than one outgoing edge, we impose time dependent distribution functions that guarantee conservation of
mass. We provide results of regularity, existence and well-posedness of the multicommodity network model for L-P-, BV- and W-1,W-p-data. Moreover, we define an L-2-tracking type objective and show
the existence of minimizers that solve the corresponding optimal control problem.}, author = {Gugat, Martin and Keimer, Alexander and Leugering, Günter and Wang, Zhiqiang}, doi = {10.3934/
nhm.2015.10.749}, faupublication = {yes}, journal = {Networks and Heterogeneous Media}, keywords = {Existence of minimizers;conservation laws;multi-commodity model;conservation laws on networks;
nonlocal conservation laws;optimal nodal control;optimal control of conservation laws on networks;systems of hyperbolic PDEs}, pages = {749-785}, peerreviewed = {Yes}, title = {{ANALYSIS} {OF} {A}
{SYSTEM} {OF} {NONLOCAL} {CONSERVATION} {LAWS} {FOR} {MULTI}-{COMMODITY} {FLOW} {ON} {NETWORKS}}, volume = {10}, year = {2015} } @article{faucris.120101344, abstract = {
There are very few results about analytic solutions of problems of optimal control with minimal L ^∞ norm. In this paper, we consider such a problem for the wave equation, where the derivative of the
state is controlled at both boundaries. We start in the zero position and consider a problem of exact control, that is, we want to reach a given terminal state in a given finite time. Our aim is to
find a control with minimal L ^∞ norm that steers the system to the target.
We give the analytic solution for certain classes of target points, for example, target points that are given by constant functions. For such targets with zero velocity, the analytic solution has
been given by Bennighof and Boucher in Ref. 1.
}, author = {Gugat, Martin}, doi = {10.1023/A:1016091803139}, faupublication = {no}, journal = {Journal of Optimization Theory and Applications}, keywords = {Optimal Control; Wave Equation; Analytic
Solution; Distributed Parameter Systems}, note = {UnivIS-Import:2015-03-09:Pub.2002.nat.dma.lama1.analyt}, pages = {397-421}, peerreviewed = {Yes}, title = {{Analytic} {Solutions} of {L}-infinity
optimal control problems for the wave equation}, volume = {114}, year = {2002} } @article{faucris.241516538, abstract = {In this paper, we propose the numerical approximation of fractional initial
and boundary value problems using Haar wavelets. In contrast to the Haar wavelet methods available in literature, where the fractional derivative of the function is approximated using the Haar basis,
we approximate the function and its classical derivatives using Haar basis functions. Moreover, error bounds in the approximation of fractional integrals and the fractional derivatives are derived,
which depend on the index J of the approximation space VJ and the fractional order α. A neural network problem modeled by a system of nonlinear fractional differential equations is also solved using
the proposed method. The numerical results show that the proposed numerical approach is efficient.}, author = {Mehandiratta, Vaibhav and Mehra, Mani and Leugering, Günter}, doi = {10.1002/mma.6800},
faupublication = {yes}, journal = {Mathematical Methods in the Applied Sciences}, keywords = {fractional calculus; fractional differential equations; Haar wavelet; square integrable functions}, note
= {CRIS-Team Scopus Importer:2020-08-14}, peerreviewed = {Yes}, title = {{An} approach based on {Haar} wavelet for the approximation of fractional calculus with application to initial and boundary
value problems}, year = {2020} } @article{faucris.122573044, abstract = {This paper considers dynamical systems under feedback with control actions limited to switching. The authors wish to
understand the closed-loop systems as approximating multi-scale problems in which the implementation of switching merely acts on a fast scale. Such hybrid dynamical systems are extensively studied in
the literature, but not much so far for feedback with partial state observation. This becomes in particular relevant when the dynamical systems are governed by partial differential equations. The
authors introduce an augmented BV setting which permits recognition of certain fast scale effects and give a corresponding well-posedness result for observations with such minimal regularity. As an
application for this setting, the authors show existence of solutions for systems of semilinear hyperbolic equations under such feedback with pointwise observations.}, author = {Hante, Falk and
Leugering, Günter and Seidman, Thomas I.}, doi = {10.1007/s11424-010-0140-0}, faupublication = {yes}, journal = {Journal of Systems Science & Complexity}, pages = {456--466}, peerreviewed = {Yes},
title = {{An} augmented {BV} setting for feedback switching control}, volume = {23}, year = {2010} } @incollection{faucris.122069024, address = {Oberwolfach}, author = {Leugering, Günter and Kocvara,
Michal and Stingl, Michael}, booktitle = {Optimal Control of Coupled Systems of PDE}, doi = {10.4171/OWR/2008/13}, editor = {Karl Kunisch, Günter Leugering, Jürgen Sprekels and Fredi Tröltzsch},
faupublication = {yes}, pages = {647-649}, peerreviewed = {unknown}, publisher = {European Mathematical Society Publishing House}, series = {Oberwolfach Reports}, title = {{A} {New} {Method} for the
{Solution} of {Multi}-{Disciplinary} {Free} {Material} {Optimization} {Problems}}, volume = {13}, year = {2008} } @incollection{faucris.107386224, abstract = {A new method and algorithm for the
efficient solution of a class of nonlinear semidefinite programming problems is introduced. The new method extends a concept proposed recently for the solution of convex semidefinite programs based
on the sequential convex programming (SCP) idea. In the core of the method, a generally non-convex semidefinite program is replaced by a sequence of subproblems, in which nonlinear constraint and
objective functions defined in matrix variables are approximated by block separable convex models. Global convergence is proved under reasonable assumptions. The article is concluded by numerical
experiments with challenging Free Material Optimization problems subject to displacement constraints.}, author = {Stingl, Michael and Kocvara, Michal and Leugering, Günter}, booktitle = {Optimal
control of coupled systems of partial differential equations}, doi = {10.1007/978-3-7643-8923-9{\_}16}, faupublication = {yes}, pages = {275--295}, peerreviewed = {unknown}, series = {Internat. Ser.
Numer. Math.}, title = {{A} {New} {Non}-linear {Semidefinite} {Programming} {Algorithm} with an {Application} to {Multidisciplinary} {Free} {Material} {Optimization}}, volume = {158}, year = {2009} }
@article{faucris.119453224, abstract = {We consider the problem of time-optimal boundary control of a one-dimensional vibrating system subject to a control constraint that prescribes an upper bound
for the L^2-norm of the image of the control function under a Volterra operator. For the solution of this problem, we propose to use Newton's method to compute the zero of the optimal value function
of certain parametric auxiliary problems, where the steering time is the parameter. The formulation of the auxiliary problems, which are problems of norm-minimal control, is based on the method of
moments. For a fixed parameter, these problems have a simple structure. We present convergence results with respect to the discretization parameters, where the discretization is done by truncating
the system of moment equations. We prove that the optimal value function of the discretized parametric auxiliary problem is differentiable and show how the derivative can be computed, so that
Newton's method can be used. We present numerical examples for the problem of time-optimal control of the rotation of an Euler-Bernoulli beam that illustrate the fast convergence of the algorithm
with respect to the time-parameter. © 2000 Elsevier Science B.V.}, author = {Gugat, Martin}, doi = {10.1016/S0377-0427(99)00291-5}, faupublication = {no}, journal = {Journal of Computational and
Applied Mathematics}, keywords = {49M99; 90C31; 93C20; Moment problems; Parametric optimization; Optimal value function; Time-optimal control; Time-parametric auxiliary problem; Rotating beam}, month
= {Jan}, note = {UnivIS-Import:2015-03-09:Pub.2000.nat.dma.zentr.anewto}, pages = {103-119}, peerreviewed = {Yes}, title = {{A} {Newton} method for the computation of time-optimal boundary controls
of one-dimensional vibrating systems}, volume = {114}, year = {2000} } @article{faucris.112462504, abstract = {For vibrating systems, a delay in the application of a feedback control may destroy the
stabilizing effect of the control. In this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end.
We show that certain delays in the boundary feedback preserve the exponential stability of the system. In particular, we show that the system is exponentially stable with delays freely switching
between the values 4Lc and 8Lc, where L is the length of the string and c is the wave speed. © 2011 Elsevier B.V. All rights reserved.}, author = {Gugat, Martin and Tucsnak, Marius}, doi = {10.1016/
j.sysconle.2011.01.004}, faupublication = {yes}, journal = {Systems & Control Letters}, keywords = {Boundary feedback; Delay; Feedback stabilization of pdes; Feedback with delay; Hyperbolic pde; Past
observation; String; Switching delay; Wave equation}, note = {UnivIS-Import:2015-03-09:Pub.2011.nat.dma.zentr.anexam}, pages = {226-230}, peerreviewed = {Yes}, title = {{An} example for the switching
delay feedback stabilization of an infinite dimensional system: {The} boundary stabilization of a string}, volume = {60}, year = {2011} } @misc{faucris.110459844, author = {Jahn, Johannes},
faupublication = {yes}, note = {UnivIS-Import:2015-03-05:Pub.1995.nat.dma.pama21.angewa}, peerreviewed = {automatic}, title = {{Angewandte} nichtlineare {Vektoroptimierung}}, year = {1995} } @article
{faucris.110282524, abstract = {In this paper an interior point method is presented for nonlinear programming problems with inequality constraints. On defining a modified distance function the
original problem is solved sequentially by using a method of feasible directions. At each iteration a usable feasible direction can be determined explicitly. Under certain assumptions it can be shown
that every accumulation point of the sequence of points constructed by the proposed algorithm satisfies the Kuhn-Tucker conditions. © 1979 Physica-Verlag.}, author = {Jahn, Johannes}, doi = {10.1007/
BF01917332}, faupublication = {no}, journal = {Mathematical Methods of Operations Research}, note = {UnivIS-Import:2015-03-05:Pub.1979.nat.dma.pama21.aninte}, pages = {1-15}, peerreviewed = {Yes},
title = {{An} {Interior} {Point} {Method} for {Nonlinear} {Programming}}, volume = {23}, year = {1979} } @article{faucris.117450344, abstract = {We consider an exact boundary control problem for the
wave equation with given initial and terminal data and Dirichlet boundary control. The aim is to steer the state of the system that is defined on a given domain to a position of rest in finite time.
The optimal control that is obtained as the solution of the problem depends on the data that define the problem, in particular on the domain. Often for the numerical solution of the control problem,
this given domain is replaced by a polygon. This is the motivation to study the convergence of the optimal controls for the polygon to the optimal controls for the given domain. To study the
convergence, the values of the optimal controls that are defined on the boundaries of the approximating polygons are mapped in the normal directions of the polygon to control functions defined on the
boundary of the original domain. This map has already been used by Bramble and King, Deckelnick, Guenther and Hinze and by Casas and Sokolowski. Using this map, we can show the strong convergence of
the transformed controls as the polygons approach the given domain. An essential tool to obtain the convergence is a regularization term in the objective functions to increase the regularity of the
state.}, author = {Gugat, Martin and Sokolowski, Jan}, doi = {10.1080/00036811.2012.724404}, faupublication = {yes}, journal = {Applicable Analysis}, keywords = {wave equation; Dirichlet boundary
control; exact control; optimal control; domain sensitivity; polygonal domain; 49J20; 35L05; 35L53}, note = {UnivIS-Import:2015-03-09:Pub.2012.nat.dma.lama1.anoteo}, pages = {2200-2214}, peerreviewed
= {Yes}, title = {{A} note on the approximation of {Dirichlet} boundary control problems for the wave equation on curved domains}, url = {http://www.tandfonline.com/doi/abs/10.1080/
00036811.2012.724404}, volume = {92}, year = {2012} } @article{faucris.255804887, abstract = {In this paper we discuss some issues related to Poincaré’s inequality for a special class of weighted
Sobolev spaces. A common feature of these spaces is that they can be naturally associated with differential operators with variable diffusion coefficients that are not uniformly elliptic. We give a
classification of these spaces in the 1-D case bases on a measure of degeneracy of the corresponding weight coefficient and study their key properties.}, author = {Kogut, Peter I. and Kupenko, Olha
P. and Leugering, Günter and Wang, Yue}, doi = {10.15421/141905}, faupublication = {yes}, journal = {Journal of Optimization, Differential Equations and Their Applications}, keywords = {Degenerate
equation; Poincaré’s inequality; Weighted Sobolev spaces}, note = {CRIS-Team Scopus Importer:2021-04-20}, pages = {1-22}, peerreviewed = {Yes}, title = {{A} {NOTE} on {WEIGHTED} {SOBOLEV} {SPACES}
{RELATED} to {WEAKLY} and {STRONGLY} {DEGENERATE} {DIFFERENTIAL} {OPERATORS}}, volume = {27}, year = {2019} } @article{faucris.119326284, abstract = {
In this paper, a new Ansatz for modelling the Baculovirus infection cycle is presented. The base of this model is the cell cycle distribution at the time of infection. It is possible to calculate the
growth of the culture and the initiation of virus processing by considering cell cycle distribution. By taking into account the length of the viral genome and the polymerase activity, it is possible
to calculate the virus production rate, which underlies a logistic growth. In the present work, a new hypothesis explaining the accelerated death rates of infected cells has been introduced. This
assumption provides the possibilities of performing calculation without any fixed time intervals. The simulation was tested by comparing experimental data with the model prediction. Therefore, cell
cycle distributions over the culture time and the growth behaviour of infected and non-infected insect cells were measured. A model, Baculovirus coding for GFP was employed for the present
investigation, as it allows tracking the infection and determining the effectiveness of the infection, which is highly dependent on the cell density at the time of infection (TOI). Furthermore, the
new model is is taken to simulate data gained from literature about virus release and adsorption. The new assumptions make the model more independent to fit into different cultivation systems.
}, author = {Lindenberger, Christoph and Pflug, Lukas and Hübner, Holger and Buchholz, Rainer}, doi = {10.1007/s12257-011-0371-5}, faupublication = {yes}, journal = {Biotechnology and Bioprocess
Engineering}, keywords = {baculovirus; modelling; insect cells; infection cycle}, pages = {211-217}, peerreviewed = {Yes}, title = {{A} novel model for studying baculovirus infection process}, url =
{http://link.springer.com/article/10.1007/s12257-011-0371-5}, volume = {17}, year = {2012} } @article{faucris.119922044, abstract = {A model comprised of a nonlinear von Karman plate coupled with a
nonlinear beam equation is developed from first principles. Dynamic junction conditions are imposed at the interface. Wellposedness is established by first considering a corresponding linear problem,
then applying a perturbation theorem for nonlinear semigroups. Proof of regularity takes advantage of elliptic theory, as well as the regularity of the Airy's stress function. The compatibility
constraints at the junction give rise to mathematical challenges not seen in earlier work on the individual plate and beam models. (C) 2005 Elsevier Ltd. All rights reserved.}, author = {Leugering,
Günter and Horn, Mary Ann}, doi = {10.1016/j.na.2005.01.048}, faupublication = {yes}, journal = {Nonlinear Analysis - Theory Methods & Applications}, keywords = {Plate-beam model;Nonlinear dynamics;
Existence and uniqueness}, pages = {E1529-E1539}, peerreviewed = {Yes}, title = {{An} overview of modelling challenges for a nonlinear plate-beam model}, url = {http://www.sciencedirect.com/science/
article/pii/S0362546X05000647}, volume = {63}, year = {2005} } @article{faucris.115613344, abstract = {In this paper we consider the behaviour of the multipliers for one-parametric semi-infinite
programs near critical parameters, where the Mangasarian-Fromovitz constraint qualification can be violated. The growth of the multipliers, when the parameter approaches such a critical parameter, is
characterized by a parametric constraint qualification which is introduced here. It is equivalent to a bound on the growth of the multipliers. © Springer-Verlag 1999.}, author = {Gugat, Martin}, doi
= {10.1007/s10107990058a}, faupublication = {no}, journal = {Mathematical Programming}, keywords = {Gauvin's theorem; Karush-Kuhn-Tucker multiplier; Mangasarian-Fromovitz constraint qualification;
Modulus of multiplier growth; Parametric nonconvex program; Semi-infinite program; Singular parameters; Uniform Mangasarian-Fromovitz condition}, note =
{UnivIS-Import:2015-03-05:Pub.1999.nat.dma.lama1.aparam}, pages = {643-653}, peerreviewed = {Yes}, title = {{A} parametric view on the {Mangasarian}-{Fromovitz} constraint qualification}, volume =
{85}, year = {1999} } @article{faucris.117897384, abstract = {We consider a boundary optimal control problem for the acoustic wave equation with a final value cost criterion. A time-domain
decomposition procedure for the corresponding optimality system is introduced, which leads to a sequence of uncoupled optimality systems of local-in-time optimal control problems. The process is
inherently parallel and is suitable for real-time control applications. Convergence of the local solutions and controls to the global ones was established in an earlier publication. In this paper, a
posteriori error estimates of the difference between the local solutions and the global one are obtained in terms of the mismatch of the nth iterates, or of successive iterates, across the
time-domain break points.}, author = {Leugering, Günter and Langnese, John E.}, doi = {10.1007/s00245-002-0750-6}, faupublication = {no}, journal = {Applied Mathematics and Optimization}, keywords =
{acoustic wave equation;optimal control;domain decomposition;a posteriori error estimates}, pages = {263-290}, peerreviewed = {Yes}, title = {{A} posteriori error estimates in time-domain
decomposition of final value optimal control of the acoustic wave equation}, url = {http://link.springer.com/article/10.1007/s00245-002-0750-6}, volume = {46}, year = {2002} } @incollection
{faucris.121142824, address = {New York}, author = {Jahn, Johannes and Krabs, W.}, booktitle = {Multicriteria Optimization in Engineering and in the Sciences}, editor = {W. Stadler}, faupublication =
{yes}, note = {UnivIS-Import:2015-04-17:Pub.1988.nat.dma.pama21.applic}, pages = {49-75}, peerreviewed = {unknown}, publisher = {Plenum Press}, title = {{Applications} of {Multicriteria}
{Optimization} in {Approximation} {Theory}}, year = {1988} } @article{faucris.123696804, abstract = {We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the
so-called weighted p-Laplacian problem. The coefficient of the -Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the control such that
there is a strictly positive lower and some upper bound. In order to handle the inherent degeneracy of the -Laplacian, we use a regularization, sometimes referred to as the -Laplacian. We derive
existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted -Laplacian, where we
approximate the nonlinearity by a bounded monotone function, parametrized by κ. Further, we discuss the asymptotic behavior of the solutions to the regularized problem on each (ϵ, κ)-level as the
parameters tend to zero and infinity, respectively.}, author = {Casas, Eduardo and Kogut, Peter I. and Leugering, Günter}, doi = {10.1137/15M1028108}, faupublication = {yes}, journal = {SIAM Journal
on Control and Optimization}, keywords = {Control in coefficients; Nonlinear Dirichlet problem; Optimal control}, pages = {1406-1422}, peerreviewed = {Yes}, title = {{Approximation} of optimal
control problems in the coefficient for the p-{Laplace} equation. {I}. {Convergence} result}, volume = {54}, year = {2016} } @article{faucris.120697544, abstract = {We consider the approximation of
semigroups e ^τA and of the functions ℓ j(τA) that appear in exponential integrators by resolvent series. The interesting fact is that the resolvent series expresses the operator functions e ^τA and
ℓ j(τA), respectively, in efficiently computable terms. This is important for semigroups, where the new approximation is different from well-known approximations by rational functions, as well as for
the application of exponential integrators, which are currently of high interest and which are usually studied in a semigroup setting on Banach spaces. The approximation of the operator functions ℓ j
(τA) in a general, strongly continuous semigroup setting has not been discussed in the literature so far, but this is crucial for an application of these integrators with unbounded operators or
bounded operators (like discretization matrices) with large norm and eigenvalues somewhere in the left half plane. © 2010 Society for Industrial and Applied Mathematics.}, author = {Gugat, Martin and
Grimm, Volker}, doi = {10.1137/090768084}, faupublication = {yes}, journal = {SIAM Journal on Numerical Analysis}, keywords = {Exponential integrator; Rational approximation; Resolvent series;
Semigroup; 65F60; 65M15; 65M22}, note = {UnivIS-Import:2015-03-09:Pub.2010.nat.dma.zentr.approx}, pages = {1826-1845}, peerreviewed = {Yes}, title = {{Approximation} of {Semigroups} and {Related}
{Operator} {Functions} by {Resolvent} {Series}}, url = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJNAAM000048000005001826000001&idtype=cvips&gifs=yes}, volume = {48}, year
= {2010} } @article{faucris.241266518, abstract = {In this review we identify a new category of methods for implementing and solving structural optimization problems that has emerged over the last 20
years, which we propose to call feature-mapping methods. The two defining aspects of these methods are that the design is parameterized by a high-level geometric description and that features are
mapped onto a non-body-fitted mesh for analysis. One motivation for using these methods is to gain better control over the geometry to, for example, facilitate imposing direct constraints on
geometric features, while avoiding issues with re-meshing. The review starts by providing some key definitions and then examines the ingredients that these methods use to map geometric features onto
a fixed mesh. One of these ingredients corresponds to the mechanism for mapping the geometry of a single feature onto a fixed analysis grid, from which an ersatz material or an immersed-boundary
approach is used for the analysis. For the former case, which we refer to as the pseudo-density approach, a test problem is formulated to investigate aspects of the material interpolation, boundary
smoothing, and numerical integration. We also review other ingredients of feature-mapping techniques, including approaches for combining features (which are required to perform topology optimization)
and methods for imposing a minimum separation distance among features. A literature review of feature-mapping methods is provided for shape optimization, combined feature/free-form optimization, and
topology optimization. Finally, we discuss potential future research directions for feature-mapping methods.}, author = {Wein, Fabian and Dunning, Peter D. and Norato, Julián A.}, doi = {10.1007/
s00158-020-02649-6}, faupublication = {yes}, journal = {Structural and Multidisciplinary Optimization}, keywords = {Feature-mapping; Fixed-grid; High-level design; Structural optimization}, note =
{CRIS-Team Scopus Importer:2020-08-07}, peerreviewed = {Yes}, title = {{A} review on feature-mapping methods for structural optimization}, year = {2020} } @article{faucris.112037244, abstract = {A
new method for the efficient solution of a class of convex semidefinite programming (SDP) problems is introduced. The method extends the sequential convex programming (SCP) concept to optimization
problems with matrix variables. The basic idea of the new method is to approximate the original optimization problem by a sequence of subproblems, in which nonlinear functions (defined in matrix
variables) are approximated by block separable convex functions. The subproblems are semidefinite programs with a favorable structure which can be efficiently solved by existing SDP software. The new
method is shown to be globally convergent. The article is concluded by a series of numerical experiments with free material optimization problems demonstrating the effectiveness of the generalized
SCP approach. © 2009 Society for Industrial and Applied Mathematics.}, author = {Stingl, Michael and Kocvara, Michal and Leugering, Günter}, doi = {10.1137/070711281}, faupublication = {yes}, journal
= {SIAM Journal on Optimization}, keywords = {Material optimization; Semidefinite programming; Sequential convex programming; Structural optimization}, note =
{UnivIS-Import:2015-03-09:Pub.2009.nat.dma.lama1.aseque}, pages = {130-155}, peerreviewed = {Yes}, title = {{A} {SEQUENTIAL} {CONVEX} {SEMIDEFINITE} {PROGRAMMING} {ALGORITHM} {WITH} {AN}
{APPLICATION} {TO} {MULTIPLE}-{LOAD} {FREE} {MATERIAL} {OPTIMIZATION}}, volume = {20}, year = {2009} } @article{faucris.122288144, abstract = {A shape-topological control of singularly perturbed
variational inequalities is considered in the abstract framework for state-constrained optimization problems. Aiming at asymptotic analysis, singular perturbation theory is applied to the
geometry-dependent objective function and results in a shape-topological derivative. This concept is illustrated analytically in a one-dimensional example problem which is controlled by an
inhomogeneity posed in a domain with moving boundary.}, author = {Leugering, Günter and Kovtunenko, Viktor}, faupublication = {yes}, journal = {Eurasian Mathematical Journal}, keywords =
{Inhomogeneity; Shape-topological control; Shape-topological derivative; Singular perturbation; State-constrained optimization; Variational inequality}, pages = {41-52}, peerreviewed = {unknown},
title = {{A} shape-topological control of variational inequalities}, url = {https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85007622311&origin=inward}, volume = {7}, year = {2016} }
@article{faucris.122289024, abstract = {We consider the shape-topological control of a singularly perturbed variational inequality. The geometry-dependent state problem that we address in this paper
concerns a heterogeneous medium with a micro-object (defect) and a macro-object (crack) modeled in two dimensions. The corresponding nonlinear optimization problem subject to inequality constraints
at the crack is considered within a general variational framework. For the reason of asymptotic analysis, singular perturbation theory is applied, resulting in the topological sensitivity of an
objective function representing the release rate of the strain energy. In the vicinity of the nonlinear crack, the antiplane strain energy release rate is expressed by means of the mode-III stress
intensity factor that is examined with respect to small defects such as microcracks, holes, and inclusions of varying stiffiness. The result of shape-topological control is useful either for arrests
or rise of crack growth.}, author = {Leugering, Günter and Kovtunenko, Viktor}, doi = {10.1137/151003209}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, keywords =
{Antiplane stress intensity factor; Crack-defect interaction; Dipole tensor; Nonlinear crack with nonpenetration; Shape-topological control; Singular perturbation; Strain energy release rate;
Topological derivative; Variational inequality}, pages = {1329-1351}, peerreviewed = {Yes}, title = {{A} shape-topological control problem for nonlinear crack-defect interaction: {The} antiplane
variational model}, volume = {54}, year = {2016} } @article{faucris.122945944, abstract = {We consider general, not necessarily convex, optimization problems with inequality constraints. We show that
the smoothed penalty algorithm generates a sequence that converges to a stationary point. In particular, we show that the algorithm provides approximations of the multipliers for the inequality
constraints. The theoretical analysis is illustrated by numerical examples for optimal control problems with pointwise state constraints and Robin boundary conditions as presented by Grossmann, Kunz,
and Meischner [C. Grossmann, H. Kunz, and R. Meischner, Elliptic control by penalty techniques with control reduction, in IFIP Advances in Information and Communication Technology, Springer-Verlag,
Berlin Heidelberg, 2009, pp. 251–267].}, author = {Gugat, Martin and Herty, Michael}, doi = {10.1080/02331934.2011.588230}, faupublication = {yes}, journal = {Optimization}, keywords = {inequality
constraints; multiplier approximations; optimal control problem; smoothed penalty function; optimization with partial differential equations; state constraints; convergence analysis; 90C30; 93C20;
90C34}, pages = {379-395}, peerreviewed = {Yes}, title = {{A} smoothed--penalty iteration for state constrained optimal control problems for partial differential equations}, volume = {62}, year =
{2011} } @article{faucris.112463164, abstract = {We study the isothermal Euler equations with friction and consider non-stationary solutions locally around a stationary subcritical state on a finite
time interval. The considered control system is a quasilinear hyperbolic system with a source term. For the corresponding initial-boundary value problem we prove the existence of a continuously
differentiable solution and present a method of boundary feedback stabilization. We introduce a Lyapunov function which is a weighted and squared H^1-norm of the difference between the nonstationary
and the stationary state. We develop boundary feedback conditions which guarantee that the Lyapunov function and the H^1-norm of the difference between the non-stationary and the stationary state
decay exponentially with time. This allows us also to prove exponential estimates for the C^0- and C^1- norm.}, author = {Hirsch-Dick, Markus and Gugat, Martin and Leugering, Günter}, doi = {10.3934/
naco.2011.1.225}, faupublication = {yes}, journal = {Numerical Algebra, Control and Optimization}, keywords = {Boundary control; C; C; Conservation law; Distributed parameter system; Exponential
stability; Feedback law; Feedback stabilization; Friction term; Gas network; H; Isothermal Euler equations; Lyapunov function; Riemann invariants}, note =
{UnivIS-Import:2015-03-09:Pub.2011.nat.dma.zentr.astric}, pages = {225-244}, peerreviewed = {Yes}, title = {{A} strict {H1}-{Lyapunov} function and feedback stabilization for the isothermal {Euler}
equations with friction}, url = {http://www.aimsciences.org/journals/displayReferences.jsp?paperID=6330}, volume = {1}, year = {2011} } @article{faucris.122291224, abstract = {We consider a thin
elastic plate with piezo patches mounted on top of it. Electrodes are located on the upper and, depending on the devices, at the lower surface of the patches. This piezo actuator is coupled to an
elastic body. We develop an asymptotic procedure to derive a two-dimensional approximation of the entire structure. As a result, we obtain an inhomogeneous fourth-order plate equation with piecewise
smooth coefficients for the vertical displacement coupled to a second-order in-plane problem. The analysis and the resulting asymptotic limits help clarifying the modeling issue concerning active
piezo devices in multidimensional smart structures. Copyright © 2012 John Wiley & Sons, Ltd.}, author = {Leugering, Günter and Nazarov, Serguei A. and Slutskij, Andrey S.}, doi = {10.1002/mma.1566},
faupublication = {yes}, journal = {Mathematical Methods in the Applied Sciences}, keywords = {anisotropic material; Asymptotic analysis; effective material; piezoelectric material; plate equation},
pages = {633-658}, peerreviewed = {Yes}, title = {{Asymptotic} analysis of {3D} thin anisotropic plates with a piezoelectric patch}, volume = {35}, year = {2012} } @article{faucris.107388424,
abstract = {We derive an asymptotic one-dimensional model of a thin piezoelectric rod by means of a dimension reduction procedure. The rod is made from a heterogeneous material with a possibly
varying cross-section and distorted ends. Asymptotically exact error estimates are derived. Representation formulas for effective moduli are established and, for a concrete piezoelectric material, we
provide an explicit form.}, author = {Leugering, Günter and Nazarov, Sergei A. and Slutskij, Andrey S.}, doi = {10.1002/zamm.201100169}, faupublication = {yes}, journal = {ZAMM - Zeitschrift für
angewandte Mathematik und Mechanik}, pages = {529--550}, peerreviewed = {Yes}, title = {{Asymptotic} analysis of 3-{D} thin piezoelectric rods}, volume = {94}, year = {2014} } @article
{faucris.265185440, abstract = {We present a 1-D model of a junction of five thin elastic rods forming the shape of a bit brace (hand drill), or, a crankshaft. The distinguishing feature of this
junction is the existence of the so-called movable elements, which are rods and knots requiring modifications of the classical asymptotic ansatze. These consist of constant longitudinal displacements
on the edges of the skeleton of the junction and affect the transmission conditions at its nodes. We provide asymptotic formulas for the displacements, stresses and elastic energy, as well as error
estimates. An exact solution of the model is given for a particular loading.}, author = {Leugering, Günter and Nazarov, Sergei A. and Slutskij, A. S. and Taskinen, Jari}, doi = {10.1002/
zamm.201900227}, faupublication = {yes}, journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik}, note = {CRIS-Team WoS Importer:2021-10-18}, peerreviewed = {Yes}, title = {{Asymptotic}
analysis of a bit brace shaped junction of thin rods}, volume = {100}, year = {2020} } @article{faucris.117899804, abstract = {The asymptotic behavior of state-constrained semilinear optimal control
problems for distributed-parameter systems with variable compact control zones is investigated. We derive conditions under which the limiting problems can be made explicit.}, author = {Leugering,
Günter and Kogut, Peter I.}, doi = {10.1007/s10957-007-9282-1}, faupublication = {yes}, journal = {Journal of Optimization Theory and Applications}, keywords = {homogenization;optimal control;state
constraints;penalized problems}, pages = {301-321}, peerreviewed = {Yes}, title = {{Asymptotic} analysis of state constrained semilinear optimal control problems}, url = {http://link.springer.com/
article/10.1007/s10957-007-9282-1}, volume = {135}, year = {2007} } @article{faucris.264589979, author = {Buttazzo, Giuseppe and Casas, Eduardo and De Teresa, Luz and Glowinski, Roland and Leugering,
Günter and Trelat, Emmanuel and Zhang, Xu}, doi = {10.1051/cocv/2021089}, faupublication = {yes}, journal = {Esaim-Control Optimisation and Calculus of Variations}, note = {CRIS-Team WoS
Importer:2021-10-01}, peerreviewed = {unknown}, title = {{A} {TRIBUTE} {TO} {PROFESSOR} {ENRIQUE} {ZUAZUA} {ON} {HIS} {60TH} {BIRTHDAY} {PREFACE}}, volume = {27}, year = {2021} } @article
{faucris.229542767, abstract = {In this paper the turnpike phenomenon is studied for problems of optimal boundary control. We consider systems that are governed by a linear $2\times 2$ hyperbolic
partial differential equation with a source term. Turnpike results are obtained for problems of optimal Dirichlet boundary control for such systems with a strongly convex objective function that
depends on the control and the boundary traces of the system states. In the problem we also allow for a convex inequality constraint. We show that asymptotically for large $T$ the influence of the
initial state becomes smaller and smaller in the sense that the $L^2$-norm of the difference between the dynamic optimal control and the stationary control that solves the corresponding static
optimal control problem remains uniformly bounded for arbitrarily large $T$. As an application, we consider gas pipeline flow.[]^[S::S]
In this paper we consider the flow through a frictionless horizontal rectangular channel that is governed by de St. Venant equations and show that the state can be controlled in finite time from a
stationary initial state to a given stationary terminal state in such a way that during this transition, the state stays in the class of C 1 functions, so in particular no shocks occur. There is no
restriction on the initial and terminal state, so in some cases it is necessary that one or both eigenvalues of the system matrix change the sign during the process. Various different cases occur:
control between subcritical states, control between supercritical states, transition from a subcritical to a supercritical state, and transition from a supercritical to a subcritical state. In the
last two cases of a control between states of a different type, one eigenvalue of the system matrix changes its sign during the process. When this happens at a boundary point during the process, it
is necessary to switch the type of boundary conditions. We show how to construct controls where at each boundary at most one such switching is necessary.}, author = {Gugat, Martin}, doi = {10.1137/
S0363012902409660}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, keywords = {De St. Venant equations; Global controllability; Subcritical states; Supercritical
states; 35L45; 35L50; 35L65; 93C20}, note = {UnivIS-Import:2015-03-09:Pub.2003.nat.dma.lama1.bounda}, pages = {1056-1070}, peerreviewed = {Yes}, title = {{Boundary} {Controllability} between {Sub}-
and {Supercritical} {Flow}}, url = {http://www.siam.org/journals/sicon/sicon.htm}, volume = {42}, year = {2003} } @article{faucris.106334624, author = {Leugering, Günter}, faupublication = {no},
journal = {JOURNAL OF INTEGRAL EQUATIONS}, month = {Jan}, pages = {157-173}, peerreviewed = {unknown}, title = {{BOUNDARY} {CONTROLLABILITY} {IN} {ONE}-{DIMENSIONAL} {LINEAR}
{THERMOVISCOELASTICITY}}, volume = {10}, year = {1985} } @article{faucris.262432149, abstract = {In this paper, we address the problem of boundary controllability for the one-dimensional nonlinear
shallow water system, describing the free surface flow of water as well as the flow under a fixed gate structure. The system of differential equations considered can be interpreted as a simplified
model of a particular type of wave energy device converter called oscillating water column. The physical requirements naturally lead to the problem of exact controllability in a prescribed region. In
particular, we use the concept of nodal profile controllability in which at a given point (the node) time-dependent profiles for the states are required to be reachable by boundary controls. By
rewriting the system into a hyperbolic system with nonlocal boundary conditions, we at first establish the semi-global classical solutions of the system, then get the local controllability and nodal
profile using a constructive method. In addition, based on this constructive process, we provide an algorithmic concept to calculate the required boundary control function for generating a solution
for solving these control problem.}, author = {Vergara-Hermosilla, G. and Leugering, Günter and Wang, Yue}, doi = {10.1051/cocv/2021076}, faupublication = {yes}, journal = {Esaim-Control Optimisation
and Calculus of Variations}, note = {CRIS-Team WoS Importer:2021-08-06}, peerreviewed = {Yes}, title = {{Boundary} controllability of a system modelling a partially immersed obstacle}, volume = {27},
year = {2021} } @article{faucris.119924244, abstract = {It is shown that finite energy states of a vibrating viscoelastic plate of the Kelvin-Voigt type are, in general, not exactly controllable by L
[2]-boundary controls. Accordingly, we present a result on approximative controllability. The method is general.}, author = {Leugering, Günter and Schmidt, E. J. P. Georg}, doi = {10.1002/
mma.1670110502}, faupublication = {no}, journal = {Mathematical Methods in the Applied Sciences}, pages = {573-586}, peerreviewed = {Yes}, title = {{BOUNDARY} {CONTROL} {OF} {A} {VIBRATING} {PLATE}
{WITH} {INTERNAL} {DAMPING}}, url = {http://onlinelibrary.wiley.com/doi/10.1002/mma.1670110502/abstract}, volume = {11}, year = {1989} } @article{faucris.117346944, abstract = {In the application of
feedback controls, a delay may appear as a perturbation caused by the computation of the controls. For vibrating systems, this delay can destroy the stabilizing effect of the control. To avoid this
problem, we consider feedback laws where a certain delay is included as a part of the control law and not as a perturbation. We consider systems that are governed by the wave equation. As a first
system, we consider a string that is fixed at one end and stabilized with a boundary feedback with constant delay at the other end. As a second example, we consider a circular string where both ends
of the string are coupled by a feedback law. For both systems, we show the exponential stability of the proposed feedback with retarded input. Moreover, for the first system, we show robustness with
respect to variations in time of the feedback parameter. The author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.2010
© The author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.}, author = {Gugat, Martin}, doi = {10.1093/imamci/dnq007},
faupublication = {yes}, journal = {IMA Journal of Mathematical Control and Information}, keywords = {boundary feedback; circular string; delay; feedback stabilization of partial differential
equations; feedback with delay; hyperbolic partial differential equation; past observation; time-dependent feedback parameter; wave equation}, note =
{UnivIS-Import:2015-03-09:Pub.2010.nat.dma.zentr.bounda}, pages = {189-203}, peerreviewed = {Yes}, title = {{Boundary} feedback stabilization by time delay for one-dimensional wave equations}, url =
{http://imamci.oxfordjournals.org/cgi/content/abstract/dnq007?ijkey=gGdKiGMBG9950h8&keytype=ref}, volume = {27}, year = {2010} } @article{faucris.247775816, abstract = {In this work we address the
problem of boundary feedback stabilization for a geometrically exact shearable beam, allowing for large deflections and rotations and small strains. The corresponding mathematical model may be
written in terms of displacements and rotations (geometrically exact beam), or intrinsic variables (intrinsic geometrically exact beam). A nonlinear transformation relates both models, allowing us to
take advantage of the fact that the latter model is a one-dimensional first-order semilinear hyperbolic system, and deduce stability properties for both models. By applying boundary feedback controls
at one end of the beam while the other end is clamped, we show that the zero steady state of the intrinsic geometrically exact beam model is locally exponentially stable for the H-1- and H-2 norms.
The proof rests on the construction of a Lyapunov function, where the theory of Bastin and Coron [Stability and Boundary Stabilization of 1-D Hyperbolic Systems, in Progr. Nonlinear Differential
Equations Appl. 88, Birkhauser/Springer, Cham, 2016] plays a crucial role. The major difficulty in applying this theory stems from the complicated nature of the nonlinearity and lower order term
where no smallness arguments apply. Using the relationship between both models, we deduce the existence of a unique solution to the geometrically exact beam model, and properties of this solution as
time goes to +infinity.}, author = {Rodriguez, Charlotte and Leugering, Günter}, doi = {10.1137/20M1340010}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, month =
{Jan}, note = {CRIS-Team WoS Importer:2021-01-15}, pages = {3533-3558}, peerreviewed = {Yes}, title = {{BOUNDARY} {FEEDBACK} {STABILIZATION} {FOR} {THE} {INTRINSIC} {GEOMETRICALLY} {EXACT} {BEAM}
{MODEL}}, volume = {58}, year = {2020} } @article{faucris.261333644, abstract = {The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear hyperbolic
equations. In this system, the influence of certain source terms that model friction effects is essential. Often for the solution of control problems it is convenient to replace the quasilinear model
by a simpler semilinear model. In this paper, we analyze the behavior of such a semilinear model on a star-shaped network. The model is derived from the diagonal form of the quasilinear model by
replacing the eigenvalues by the sound speed multiplied by 1 or -1 respectively. Thus in the corresponding eigenvalues the influence of the gas velocity is neglected, which is justified in the
applications since it is much smaller than the sound speed in the gas. For a star-shaped network of horizontal pipes for suitable coupling conditions we present boundary feedback laws that stabilize
the system state exponentially fast to a position of rest for sufficiently small initial data. We show the exponential decay of the L2-norm for arbitrarily long pipes. This is remarkable since in
general even for linear systems, for certain source terms the system can become exponentially unstable if the space interval is too long. Our proofs are based upon an observability inequality and
suitably chosen Lyapunov functions. At the end of the paper, numerical examples are presented that include a comparison of the semilinear model and the quasilinear system. }, author = {Gugat, Martin
and Giesselmann, Jan}, doi = {10.1051/cocv/2021061}, faupublication = {yes}, journal = {Esaim-Control Optimisation and Calculus of Variations}, keywords = {Dirichlet feedback; Exponential stability;
Isothermal Euler equations; Observability inequality; Pipeline networks; Real gas; Riemann invariants; Semilinear model; Source terms; Stabilization}, note = {CRIS-Team Scopus Importer:2021-07-09},
peerreviewed = {Yes}, title = {{Boundary} feedback stabilization of a semilinear model for the flow in star-shaped gas networks}, volume = {27}, year = {2021} } @article{faucris.114066964, abstract =
{The Schlögl system is governed by a nonlinear reaction-diffusion partial differential equation with a cubic nonlinearity that determines three constant equilibrium states. It is a classical example
of a chemical reaction system that is bistable. The constant equilibrium that is enclosed by the other two constant equilibrium points is unstable. In this paper, Robin boundary feedback laws are
presented that stabilize the system in a given stationary state or more generally in a given time-dependent desired system orbit. The exponential stability of the closed loop system with respect to
the ^L2-norm is proved. In particular, it is shown that with the boundary feedback law the unstable constant equilibrium point can be stabilized.}, author = {Gugat, Martin and Tröltzsch, Fredi}, doi
= {10.1016/j.automatica.2014.10.106}, faupublication = {yes}, journal = {Automatica}, keywords = {Boundary feedback; Exponential stability; Lyapunov function; Parabolic partial differential equation;
Periodic operation; Poincaré-Friedrichs inequality; Robin feedback; Stabilization of desired orbits; Stabilization of periodic orbits}, month = {Jan}, note =
{UnivIS-Import:2015-03-09:Pub.2015.nat.dma.zentr.bounda}, pages = {192-199}, peerreviewed = {Yes}, title = {{Boundary} feedback stabilization of the {Schlögl} system}, volume = {51}, year = {2015} }
@article{faucris.113899544, abstract = {We study a semilinear mildly damped wave equation that contains the telegraph equation as a special case. We consider Neumann velocity boundary feedback and
prove the exponential stability of the closed loop system. We show that for vanishing damping term in the partial differential equation, the decay rate of the system approaches the rate for the
system governed by the wave equation without damping term. In particular, this implies that arbitrarily large decay rates can occur if the velocity damping in the partial differential equation is
sufficiently small. © 2014 Elsevier B.V. All rights reserved.}, author = {Gugat, Martin}, doi = {10.1016/j.sysconle.2014.01.007}, faupublication = {yes}, journal = {Systems & Control Letters},
keywords = {Anti-damping; Boundary feedback; Damping; Decay rate; Exponential stability; Hyperbolic partial differential equation; Nonlinear wave equation; Semilinear wave equation; Telegraph
equation}, note = {UnivIS-Import:2015-03-09:Pub.2014.nat.dma.lama1.bounda}, pages = {72-84}, peerreviewed = {Yes}, title = {{Boundary} feedback stabilization of the telegraph equation: {Decay} rates
for vanishing damping term}, url = {http://ac.els-cdn.com/S0167691114000279/1-s2.0-S0167691114000279-main.pdf?{\_}tid=631b42fa-9efc-11e3-83ef-00000aab0f26&acdnat=1393429438{\_}
c7a54e1446ab014020ff9d6d7536964a}, volume = {66}, year = {2014} } @article{faucris.201253582, author = {Gugat, Martin and Rosier, Lionel and Perrolaz, Vincent}, doi = {10.1007/s00028-018-0449-z},
faupublication = {yes}, journal = {Journal of Evolution Equations}, peerreviewed = {Yes}, title = {{Boundary} stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for
small source terms}, url = {https://link.springer.com/article/10.1007/s00028-018-0449-z}, year = {2018} } @inproceedings{faucris.121010164, author = {Rathmann, Wigand and Vogel, Frank and Landes,
Hermann and Kaltenbacher, Manfred}, booktitle = {21. CAD-FEM Users' Meeting}, faupublication = {no}, peerreviewed = {No}, publisher = {CAD-FEM GmbH Grafing}, title = {{Calculation} of {Sound}
{Fields} in {Flowing} {Media} {Using} {CAPA} and {Diffpack}}, year = {2003} } @incollection{faucris.111139424, abstract = {We consider optimal control problems for the flow of gas or fresh water in
pipe networks as well as drainage or sewer systems in open canals. The equations of motion are taken to be represented by the nonlinear isothermal Euler gas equations, the water hammer equations, or
the St. Venant equations for flow. We formulate model hierarchies and derive an abstract model for such network flow problems including pipes, junctions, and controllable elements such as valves,
weirs, pumps, as well as compressors. We use the abstract model to give an overview of the known results and challenges concerning equilibria, well-posedness, controllability, and optimal control. A
major challenge concerning the optimization is to deal with switching on-off states that are inherent to controllable devices in such applications combined with continuous simulation and optimization
of the gas flow. We formulate the corresponding mixed-integer nonlinear optimal control problems and outline a decomposition approach as a solution technique.}, address = {Singapore}, author =
{Hante, Falk and Leugering, Günter and Martin, Alexander and Schewe, Lars and Schmidt, Martin}, booktitle = {Industrial Mathematics and Complex Systems: Emerging Mathematical Models, Methods and
Algorithms}, doi = {10.1007/978-981-10-3758-0{\_}5}, editor = {Manchanda, Pammy; Lozi, René; Siddiqi, Abul Hasan}, faupublication = {yes}, isbn = {978-981-10-3758-0}, keywords = {Networks, pipes,
canals, Euler and St. Venant equations, hierarchy of models, domain decomposition, controllability, optimal control.}, pages = {77-122}, peerreviewed = {unknown}, publisher = {Springer Singapore},
series = {Industrial and Applied Mathematics}, title = {{Challenges} in {Optimal} {Control} {Problems} for {Gas} and {Fluid} {Flow} in {Networks} of {Pipes} and {Canals}: {From} {Modeling} to
{Industrial} {Applications}}, url = {https://opus4.kobv.de/opus4-trr154/files/121/isiam-paper.pdf}, year = {2017} } @article{faucris.112244264, abstract = {We consider the subcritical flow in gas
networks consisting of a finite linear sequence of pipes coupled by compressor stations. Such networks are important for the transportation of natural gas over large distances to ensure sustained gas
supply. We analyse the system dynamics in terms of Riemann invariants and study stationary solutions as well as classical non-stationary solutions for a given finite time interval. Furthermore, we
construct feedback laws to stabilize the system locally around a given stationary state. To do so we use a Lyapunov function and prove exponential decay with respect to the L^2-norm. © American
Institute of Mathematical Sciences.}, author = {Hirsch-Dick, Markus and Gugat, Martin and Leugering, Günter}, doi = {10.3934/nhm.2010.5.691}, faupublication = {yes}, journal = {Networks and
Heterogeneous Media}, keywords = {Classical solutions; Critical length; Feedback law; Gas networks; Gun barrel; Linear networks; Lyapunov function; Networked hyperbolic systems; Riemann invariants},
note = {UnivIS-Import:2015-03-09:Pub.2010.nat.dma.zentr.classi}, pages = {691-709}, peerreviewed = {Yes}, title = {{Classical} solutions and feedback stabilization for the gas flow in a sequence of
pipes}, url = {http://aimsciences.org/journals/displayArticles.jsp?paperID=5645}, volume = {5}, year = {2010} } @article{faucris.112749824, abstract = {Compressible squeeze film damping is a
phenomenon of great importance for micromachines. For example, for the optimal design of an electrostatically actuated micro-cantilever mass sensor that operates in air, it is essential to have a
model for the system behavior that can be evaluated efficiently. An analytical model that is based upon a solution of the linearized Reynolds equation has been given by R.B. Darling. In this paper we
explain how some infinite sums that appear in Darling’s model can be evaluated analytically. As an example of applications of these closed form representations, we compute an approximation for the
critical frequency where the spring component of the reaction force on the microplate, due to the motion through the air, is equal to a certain given multiple of the damping component. We also show
how some double series that appear in the model can be reduced to a single infinite series that can be approximated efficiently.}, author = {Gugat, Martin}, doi = {10.3390/app2020479}, faupublication
= {yes}, journal = {Applied Sciences}, keywords = {squeeze film; optimal sensor design; resonant mass sensor; double series; efficient approximation of double series; evaluation of infinite series;
Darling’s model}, note = {UnivIS-Import:2015-03-09:Pub.2012.nat.dma.zentr.closed}, pages = {479-484}, peerreviewed = {unknown}, title = {{Closed} {Form} {Representations} of {Some} {Series} in
{Darling} s {Model} for {Squeeze} {Film} {Damping} with a {Rectangular} {Plate}}, url = {http://www.mdpi.com/2076-3417/2/2/479}, volume = {2}, year = {2012} } @misc{faucris.118869344, abstract = {We
consider nonlinear and nonsmooth mixing aspects in gas transport optimization problems. As mixed-integer reformulations of pooling-type mixing models already render small-size instances
computationally intractable, we investigate the applicability of smooth nonlinear programming techniques for equivalent complementarity-based reformulations. Based on recent results for remodeling
piecewise affine constraints using an inverse parametric quadratic programming approach, we show that classical stationarity concepts are meaningful for the resulting complementarity-based
reformulation of the mixing equations. Further, we investigate in a numerical study the performance of this reformulation compared to a more compact complementarity-based one that does not feature
such beneficial regularity properties. All computations are performed on publicly available data of real-world size problem instances from steady-state gas transport.}, author = {Hante, Falk and
Schmidt, Martin}, faupublication = {yes}, keywords = {Gas transport networks, Mixing, Inverse parametric quadratic programming, Complementarity constraints, MPCC}, peerreviewed = {automatic}, title =
{{Complementarity}-{Based} {Nonlinear} {Programming} {Techniques} for {Optimal} {Mixing} in {Gas} {Networks}}, url = {http://www.optimization-online.org/DB{\_}HTML/2017/09/6198.html}, year = {2017} }
@article{faucris.220859888, abstract = {We consider nonlinear and nonsmooth mixing aspects in gas transport optimization problems. As mixed-integer reformulations of pooling-type mixing models
already render small-size instances computationally intractable, we investigate the applicability of smooth nonlinear programming techniques for equivalent complementarity-based reformulations. Based
on recent results for remodeling piecewise affine constraints using an inverse parametric quadratic programming approach, we show that classical stationarity concepts are meaningful for the resulting
complementarity-based reformulation of the mixing equations. Further, we investigate in a numerical study the performance of this reformulation compared to a more compact complementarity-based one
that does not feature such beneficial regularity properties. All computations are performed on publicly available data of real-world size problem instances from steady-state gas transport.}, author =
{Hante, Falk and Schmidt, Martin}, doi = {10.1007/s13675-019-00112-w}, faupublication = {yes}, journal = {EURO Journal on Computational Optimization}, keywords = {Complementarity constraints; Gas
transport networks; Inverse parametric quadratic programming; Mixing; MPCC}, note = {CRIS-Team Scopus Importer:2019-06-18}, peerreviewed = {Yes}, title = {{Complementarity}-based nonlinear
programming techniques for optimal mixing in gas networks}, year = {2019} } @article{faucris.111644544, author = {Knossalla, Martin}, faupublication = {yes}, journal = {Journal of Nonlinear and
Variational Analysis}, pages = {265-279}, peerreviewed = {Yes}, title = {{Concepts} on generalized epsilon-subdifferentials for minimizing locally {Lipschitz} continuous functions}, url = {http://
jnva.biemdas.com/archives/404}, year = {2017} } @article{faucris.116673304, abstract = {
We consider models based on conservation laws. For the optimization of
such systems, a sensitivity analysis is essential to determine how changes in the decision
variables influence the objective function. Here we study the sensitivity with respect
to the initial data of objective functions that depend upon the solution of Riemann
problems with piecewise linear flux functions. We present representations for the one–
sided directional derivatives of the objective functions. The results can be used in the
numerical method called Front-Tracking.
}, author = {Gugat, Martin and Herty, Michael and Klar, Axel and Leugering, Günter}, doi = {10.1051/m2an:2006037}, faupublication = {yes}, journal = {Esaim-Mathematical Modelling and Numerical
Analysis-Modelisation Mathematique Et Analyse Numerique}, keywords = {Sensitivity calculus; front-tracking; conservation laws; 35Lxx; 76N15}, month = {Jan}, note =
{UnivIS-Import:2015-03-09:Pub.2007.nat.dma.zentr.conser}, pages = {939-960}, peerreviewed = {Yes}, title = {{Conservation} {Law} {Constrained} {Optimization}}, url = {http://www.edpsciences.org/
10.1051/m2an:2006037}, volume = {40}, year = {2007} } @article{faucris.123262964, abstract = {While the numerical discretization of one-dimensional blood flow models for vessels with viscoelastic
wall properties is widely established, there is still no clear approach on how to couple one-dimensional segments that compose a network of viscoelastic vessels. In particular for Voigt-type
viscoelastic models, assumptions with regard to boundary conditions have to be made, which normally result in neglecting the viscoelastic effect at the edge of vessels. Here we propose a coupling
strategy that takes advantage of a hyperbolic reformulation of the original model and the inherent information of the resulting system. We show that applying proper coupling conditions is fundamental
for preserving the physical coherence and numerical accuracy of the solution in both academic and physiologically relevant cases. (C) 2016 Elsevier Inc. All rights reserved.}, author = {Müller, Lukas
and Leugering, Günter and Blanco, Pablo J.}, doi = {10.1016/j.jcp.2016.03.012}, faupublication = {yes}, journal = {Journal of Computational Physics}, keywords = {Finite volume schemes;
Viscoelasticity;Junctions;One-dimensional blood flow}, pages = {167-193}, peerreviewed = {Yes}, title = {{Consistent} treatment of viscoelastic effects at junctions in one-dimensional blood flow
models}, url = {http://www.sciencedirect.com/science/article/pii/S0021999116001649}, volume = {314}, year = {2016} } @book{faucris.118439244, abstract = {This special volume focuses on optimization
and control of processes governed by partial differential equations. The contributors are mostly participants of the DFG-priority program 1253: Optimization with PDE-constraints which is active since
2006. The book is organized in sections which cover almost the entire spectrum of modern research in this emerging field. Indeed, even though the field of optimal control and optimization for
PDE-constrained problems has undergone a dramatic increase of interest during the last four decades, a full theory for nonlinear problems is still lacking. The contributions of this volume, some of
which have the character of survey articles, therefore, aim at creating and developing further new ideas for optimization, control and corresponding numerical simulations of systems of possibly
coupled nonlinear partial differential equations. The research conducted within this unique network of groups in more than fifteen German universities focuses on novel methods of optimization,
control and identification for problems in infinite-dimensional spaces, shape and topology problems, model reduction and adaptivity, discretization concepts and important applications. Besides the
theoretical interest, the most prominent question is about the effectiveness of model-based numerical optimization methods for PDEs versus a black-box approach that uses existing codes, often
heuristic-based, for optimization.}, address = {Basel}, doi = {10.1007/978-3-0348-0133-1}, edition = {1}, editor = {Leugering, Günter and Engell, Sebastian and Griewank, Andreas and Hinze, Michael
and Rannacher, Rolf and Schulz, Volker and Ulbrich, Michael and Ulbrich, Stefan}, faupublication = {yes}, isbn = {978-3-0348-0132-4}, note = {UnivIS-Import:2015-05-08:Pub.2012.nat.dma.lama1.wellpo},
publisher = {Birkhäuser}, series = {ISNM}, title = {{Constrained} {Optimization} and {Optimal} {Control} for {Partial} {Differential} {Equations}}, url = {http://www.springer.com/de/book/
9783034801324}, volume = {160}, year = {2012} } @article{faucris.112679424, abstract = {We consider a water distribution network where at a finite number of nodes, contaminant injection can occur. We
consider the problem of the identification of the contaminations. This problem can be considered as an optimal control problem with a networked system that is governed by a transport reaction
equation. The identification is based upon observations from a finite number of sensors. The corresponding infinite-dimensional optimization problem is defined in a Hilbert space setting. In order to
guarantee that our optimization problem has a unique solution, a quadratic regularization term is added in the objective function. Under certain assumptions on the relations of the travel times
through the pipes we obtain a representation of this optimization problem that allows the computation of the solution on a discrete time grid by solving finite-dimensional linear least squares
problems. On these time grids, there is no discretization error since our approach is based upon an exact representation of the system state. This is useful to minimize potential impacts of
contamination emergencies on consumers by helping to select locations to flush the contaminant out of the distribution network. © 2012 Society for Industrial and Applied Mathematics.}, author =
{Gugat, Martin}, doi = {10.1137/110859269}, faupublication = {yes}, journal = {SIAM Journal on Applied Mathematics}, keywords = {Contamination emergency; Hyperbolic PDE; Identification; Least
squares; Network; Network dynamics; Observation; Optimal control; PDE-constrained optimization; Transport reaction equation}, note = {UnivIS-Import:2015-03-09:Pub.2012.nat.dma.lama1.contam}, pages =
{1772-1791}, peerreviewed = {Yes}, title = {{Contamination} {Source} {Determination} in {Water} {Distribution} {Networks}}, url = {http://epubs.siam.org/doi/abs/10.1137/110859269}, volume = {72},
year = {2012} } @article{faucris.115449004, abstract = {In this paper we introduce the concept of the contingent epiderivative for a set-valued map which modifies a notion introduced by Aubin [2] as
upper contingent derivative. It is shown that this kind of a derivative has important properties and is one possible generalization of directional derivatives in the single-valued convex case. For
optimization problems with a set-valued objective function optimality conditions based on the concept of the contingent epiderivative are proved which are necessary and sufficient under suitable
assumptions.}, author = {Jahn, Johannes and Rauh, R.}, faupublication = {yes}, journal = {Mathematical Methods of Operations Research}, keywords = {Convex and set-valued analysis; Optimality
conditions; Vector optimization}, note = {UnivIS-Import:2015-03-05:Pub.1997.nat.dma.pama21.contin}, pages = {193-211}, peerreviewed = {Yes}, title = {{Contingent} {Epiderivatives} and {Set}-{Valued}
{Optimization}}, volume = {46}, year = {1997} } @article{faucris.119092424, author = {Knossalla, Martin}, doi = {10.1007/s11228-018-0481-8}, faupublication = {yes}, journal = {Set-Valued and
Variational Analysis}, pages = {1-28}, peerreviewed = {Yes}, title = {{Continuous} outer subdifferentials in nonsmooth optimization}, url = {https://link.springer.com/article/10.1007/
s11228-018-0481-8}, year = {2018} } @book{faucris.285705230, author = {Cindea, Nicolae and Leugering, Günter and Loheac, Jerome and Micu, Sorin and Iacob, Gheorghe Mihoc-Caius and Morris, Kirsten and
Takahashi, Takeo and Zuazua Iriondo, Enrique}, faupublication = {yes}, month = {Jan}, note = {CRIS-Team WoS Importer:2022-11-25}, pages = {1-2}, peerreviewed = {unknown}, title = {{CONTROL} {AND}
{ANALYSIS} {OF} {PARTIAL} {DIFFERENTIAL} {EQUATIONS}}, volume = {24}, year = {2022} } @article{faucris.107390624, abstract = {We study a wave equation in one space dimension with a general diffusion
coefficient which degenerates on part of the boundary. Degeneracy is measured by a real parameter mu(a) > 0. We establish observability inequalities for weakly (when mu(a) is an element of [0, 1[) as
well as strongly (when mu(a) is an element of [1, 2[) degenerate equations. We also prove a negative result when the diffusion coefficient degenerates too violently (i.e., when mu(a) > 2) and prove
the blowup of the observability time when mu(a) converges to 2 from below. Thus, using the Hilbert uniqueness method we deduce the exact controllability of the corresponding degenerate control
problem when mu(a) is an element of [0, 2[. We conclude the paper by studying the boundary stabilization of the degenerate linearly damped wave equation and show that a suitable boundary feedback
stabilizes the system exponentially. We extend this stability analysis to the degenerate nonlinearly boundary-damped wave equation for an arbitrarily growing nonlinear feedback close to the origin.
This analysis proves that the degeneracy does not affect the optimal energy decay rates at large time. We apply the optimal-weight convexity method of [F. Alabau-Boussouira, Appl. Math. Optim., 51
(2005), pp. 61-105], [F. Alabau-Boussouira, J. Differential Equations, 249 (2010), pp. 1473-1517], together with our results for linear damping, to perform this stability analysis.}, author =
{Leugering, Günter and Cannarsa, Piermarco and Alabau-Boussouira, Fatiha}, doi = {10.1137/15M1020538}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, keywords =
{degenerate wave equations;controllability;stabilization;boundary control}, month = {Jan}, pages = {2052-2087}, peerreviewed = {Yes}, title = {{CONTROL} {AND} {STABILIZATION} {OF} {DEGENERATE} {WAVE}
{EQUATIONS}}, volume = {55}, year = {2017} } @article{faucris.119915444, abstract = {Consider a Timoshenko beam that is clamped to an axis perpendicular to the axis of the beam. We study the problem
to move the beam from a given initial state to a position of rest, where the movement is controlled by the angular acceleration of the axis to which the beam is clamped. We show that this problem of
controllability is solvable if the time of rotation is long enough and a certain parameter that describes the material of the beam is a rational number that has an even numerator and an odd
denominator or vice versa.}, author = {Gugat, Martin}, doi = {10.1051/cocv:2001113}, faupublication = {no}, journal = {Esaim-Control Optimisation and Calculus of Variations}, keywords = {Eigenvalues;
Exact controllability; Moment problem; Rotating Timoshenko beam; 93C20; 93B05; 93B60}, note = {UnivIS-Import:2015-03-09:Pub.2001.nat.dma.lama1.contro}, pages = {333-360}, peerreviewed = {Yes}, title
= {{Controllability} of a slowly rotating {Timoshenko} beam}, url = {http://www.esaim-cocv.org/articles/cocv/abs/2001/01/cocvVol6-13/cocvVol6-13.html}, volume = {6}, year = {2001} } @article
{faucris.279554796, abstract = {In this article, we consider a system consisting of two elastic strings with attached tip masses coupled through an elastic spring. Our aim is to analyze its exact
boundary controllability properties and to characterize the spaces of controllable initial data depending on the number of controls acting on the boundaries of the strings. We show that singularities
in waves are "smoothed by three orders" as they crass a point mass. Consequently, when only one control acts on the extremity of the first string, the space of controlled initial data is asymmetric,
the components corresponding to the second string having to be more regular than those corresponding to the first one. Roughly speaking, if the initial data for the string which is directly
controlled can be in L-2 x H-1, they should be at least in H-3 x H-2 for the second string, located on the other part of the masses.}, author = {Leugering, Günter and Micu, Sorin and Roventa, Ionel
and Wang, Yue}, doi = {10.1007/s00028-022-00823-5}, faupublication = {yes}, journal = {Journal of Evolution Equations}, note = {CRIS-Team WoS Importer:2022-08-05}, peerreviewed = {Yes}, title =
{{Controllability} properties of a hyperbolic system with dynamic boundary conditions}, volume = {22}, year = {2022} } @article{faucris.109217724, abstract = {An elastic body weakened by small cracks
is considered in the framework of unilateral variational problems in linearized elasticity. The frictionless contact conditions are prescribed on the crack lips in two spatial dimensions, or on the
crack faces in three spatial dimensions. The weak solutions of the equilibrium boundary value problem for the elasticity problem are determined by minimization of the energy functional over the cone
of admissible displacements. The associated elastic energy functional evaluated for the weak solutions is considered for the purpose of control of crack propagation. The singularities of the elastic
displacement field at the crack front are characterized by the shape derivatives of the elastic energy with respect to the crack shape within the Griffith theory. The first order shape derivative of
the elastic energy functional with respect to the crack shape, i.e., evaluated for a deformation field supported in an open neighbourhood of one of crack tips, is called the Griffith functional. The
control of the crack front in the elastic body is performed by the optimum shape design technique. The Griffith functional is minimized with respect to the shape and the location of small inclusions
in the body. The inclusions are located far from the crack. In order to minimize the Griffith functional over an admissible family of inclusions, the second order directional, mixed shape-topological
derivatives of the elastic energy functional are evaluated. The domain decomposition technique is applied to the shape and topological sensitivity analysis of variational inequalities. The nonlinear
crack model in the framework of linear elasticity is considered in two and three spatial dimensions. The boundary value problem for the elastic displacement field takes the form of a variational
inequality over the positive cone in a fractional Sobolev space. The variational inequality leads to a problem of metric projection over a polyhedric convex cone, so the concept of conical
differentiability applies to shape and topological sensitivity analysis of variational inequalities under consideration.}, author = {Leugering, Günter and Sokolowski, Jan and Zochowski, Antoni}, doi
= {10.3934/dcds.2015.35.2625}, faupublication = {yes}, journal = {Discrete and Continuous Dynamical Systems}, pages = {2625--2657}, peerreviewed = {Yes}, title = {{Control} of crack propagation by
shape-topological optimization}, volume = {35}, year = {2015} } @article{faucris.117900024, abstract = {The present study is concerned with the questions of controllability and stabilizability of
planar networks of vibrating beams consisting of several Timoshenko beams connected to each other by rigid joints at all interior nodes of the system. Some of the exterior nodes are either clamped or
free; controls may be applied at the remaining exterior nodes and/or at interior joints in the form of forces and/or bending moments. For a given configuration, is it at all possible to drive all
vibrations to the rest configuration in a given finite time interval by means of controls acting at some or all of the available (nonclamped) nodes of the network and, if so, where should such
controls be placed? Alternatively, a control objective is to construct energy absorbing boundary-feedback controls that will guarantee uniform energy decay. It is demonstrated that if such a network
does not contain closed loops and if at most one of the exterior nodes is clamped, exact controllability and uniform stabilizability of the network is indeed possible by means of controls placed at
the free exterior nodes of the system. On the other hand, examples are presented to demonstrate that when a closed loop is present in the network or if the network has more than one clamped exterior
node, it may happen that approximate control of the network to its rest configuration is not possible even if controls are placed at every available node of the system.}, author = {Langnese, John E.
and Leugering, Günter and Schmidt, E. J. P. Georg}, doi = {10.1137/0331035}, faupublication = {no}, journal = {SIAM Journal on Control and Optimization}, keywords = {CONTROL OF NETWORKS;TIMOSHENKO
BEAMS;EXACT CONTROLLABILITY;UNIFORM STABILIZABILITY; 93C20; 93D15; 35B45}, pages = {780-811}, peerreviewed = {Yes}, title = {{CONTROL} {OF} {PLANAR} {NETWORKS} {OF} {TIMOSHENKO} {BEAMS}}, volume =
{31}, year = {1993} } @article{faucris.288273167, abstract = {This article concentrates on singularly perturbed static convection–diffusion equations with varying coefficients on a metric graph G=
(V,E). Our interest is in the convection dominated situation which is described by a small parameter ϵ>0 in front of the diffusion term. As ϵ→0, the reduced problem may exhibit boundary layers at the
multiple vertices as well as at the simple nodes. We analyze the possible scenarios and validate the results in several test cases. We investigate several exemplary graphs and use an upwind finite
difference method on a piece-wise Shishkin mesh. Error estimates are also discussed to show ϵ-uniform convergence.}, author = {Kumar, Vivek and Leugering, Günter}, doi = {10.1016/j.cam.2023.115062},
faupublication = {yes}, journal = {Journal of Computational and Applied Mathematics}, keywords = {Convection diffusion problem; Metric graph; Shishkin meshes; Upwind finite difference schemes}, note
= {CRIS-Team Scopus Importer:2023-01-27}, peerreviewed = {Yes}, title = {{Convection} dominated singularly perturbed problems on a metric graph}, volume = {425}, year = {2023} } @article
{faucris.255840653, abstract = {We consider a direct approach to solving the mixedinteger nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary
differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient
conditions for the convergence of this approach in the sense of the corresponding optimal values. The results are obtained by considering the discretized problem as a parametric mixed-integer
nonlinear optimization problem in finite dimensions, where the step size for discretization of the dynamics is the parameter. In this setting, we prove the continuity of the optimal value function
under a stability assumption for the integer feasible set and second-order conditions from nonlinear optimization. We address the necessity of the conditions on the example of pipe sizing problems
for gas networks.}, author = {Hante, Falk and Schmidt, Martin}, faupublication = {yes}, journal = {Control and Cybernetics}, keywords = {Lipschitz continuity; Mixedinteger nonlinear programming;
Optimal value function; Optimization with differential equations; Parametric optimization}, note = {CRIS-Team Scopus Importer:2021-04-20}, pages = {209-230}, peerreviewed = {Yes}, title =
{{Convergence} of finite-dimensional approximations for mixed-integer optimization with differential equations∗}, volume = {48}, year = {2019} } @article{faucris.111753444, author = {Hante, Falk and
Sigalotti, Mario}, doi = {10.1137/100801561}, faupublication = {no}, journal = {SIAM Journal on Control and Optimization}, pages = {752--770}, peerreviewed = {Yes}, title = {{Converse} {Lyapunov}
theorems for switched systems in {Banach} and {Hilbert} spaces}, volume = {49}, year = {2011} } @article{faucris.239156059, abstract = {In this paper, we consider a stationary model for the flow
through a network. The flow is determined by the values at the boundary nodes of the network. We call these values the loads of the network. In the applications, the feasible loads must satisfy some
box constraints. We analyze the structure of the set of feasible loads. Our analysis is motivated by gas pipeline flows, where the box constraints are pressure bounds.}, author = {Gugat, Martin and
Schultz, Ruediger and Schuster, Michael}, doi = {10.3934/nhm.2020008}, faupublication = {yes}, journal = {Networks and Heterogeneous Media}, note = {CRIS-Team WoS Importer:2020-06-09}, pages =
{171-195}, peerreviewed = {Yes}, title = {{CONVEXITY} {AND} {STARSHAPEDNESS} {OF} {FEASIBLE} {SETS} {IN} {STATIONARY} {FLOW} {NETWORKS}}, volume = {15}, year = {2020} } @article{faucris.222493071,
author = {Meßner, Arthur and Engel, Ulf}, faupublication = {yes}, journal = {Draht}, note = {LFT Import::2019-07-16 (1987)}, pages = {30-35}, peerreviewed = {No}, title = {{Das} {Werkstoffverhalten}
beim {Umformen} von {Kleinstteilen}}, volume = {48}, year = {1997} } @article{faucris.122292544, abstract = {We propose a model for a two-dimensional elastic body with a thin elastic inclusion
modeled by a beam equation. Moreover, we assume that a delamination of the inclusion may take place resulting in a crack. Nonlinear boundary conditions are imposed at the crack faces to prevent
mutual penetration between the faces. Both variational and differential problem formulations are considered, and existence of solutions is established. Furthermore, we study the dependence of the
solution on the rigidity of the embedded beam. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain a rigid beam inclusion and cracks with nonpenetration
conditions, respectively. Anisotropic behavior of the beam is also analyzed.}, author = {Leugering, Günter and Khludnev, Alexander}, doi = {10.2140/memocs.2014.2.1}, faupublication = {yes}, journal =
{Mathematics and Mechanics of Complex Systems}, keywords = {Crack; Nonlinear boundary conditions; Nonpenetration; Thin inclusion; Variational inequality}, pages = {1-21}, peerreviewed = {unknown},
title = {{Delaminated} thin elastic inclusions inside elastic bodies}, volume = {2}, year = {2014} } @article{faucris.123598024, abstract = {We present a novel approach for gradient based
maximization of phononic band gaps. The approach is a geometry projection method combining parametric shape optimization with density based topology optimization. By this approach, we obtain, in a
two dimension setting, cellular structures exhibiting relative and normalized band gaps of more than 8 and 1.6, respectively. The controlling parameter is the minimal strut size, which also
corresponds with the obtained stiffness of the structure. The resulting design principle is manually interpreted into a three dimensional structure from which cellular metal samples are fabricated by
selective electron beam melting. Frequency response diagrams experimentally verify the numerically determined phononic band gaps of the structures. The resulting structures have band gaps down to the
audible frequency range, qualifying the structures for an application in noise isolation.}, author = {Wormser, Maximilian and Wein, Fabian and Stingl, Michael and Körner, Carolin}, doi = {10.3390/
ma10101125}, faupublication = {yes}, journal = {Materials}, keywords = {phononic band gap; additive manufacturing; selective electron beam melting; cellular materials; metamaterials; topology
optimization; parametric shape optimization; gradient based optimization}, peerreviewed = {Yes}, title = {{Design} and {Additive} {Manufacturing} of {3D} {Phononic} {Band} {Gap} {Structures} {Based}
on {Gradient} {Based} {Optimization}}, volume = {10}, year = {2017} } @article{faucris.107381384, abstract = {The optimization and manufacturing of an auxetic structure is presented. An inverse
homogenization method is used to obtain the optimized geometry shown in the figure. The resulting structure is then produced using selective electron beam melting. The numerically predicted
properties are experimentally verifie}, author = {Schwerdtfeger, Jan and Wein, Fabian and Leugering, Günter and Singer, Robert and Körner, Carolin and Stingl, Michael and Schury, F.}, doi = {10.1002/
adma.201004090}, faupublication = {yes}, journal = {Advanced Materials}, pages = {2650--2654}, peerreviewed = {Yes}, title = {{Design} of {Auxetic} {Structures} via {Mathematical} {Optimization}},
volume = {23}, year = {2011} } @inproceedings{faucris.223919809, abstract = {None}, author = {Geiger, Manfred and Meßner, Arthur and Engel, Ulf and Kals, Roland and Vollertsen, Frank}, booktitle =
{Chipless 2000 Advancing Chipless Component Manufacture, Proceedings of the 9th International Cold Forging Congress}, editor = {Standring, P.}, faupublication = {yes}, note = {LFT Import::2019-08-05
(1971)}, pages = {155-164}, peerreviewed = {unknown}, title = {{Design} of {Micro} {Forming} {Proces}¬ses {Fundamentals}, {Mate}¬rial {Data} and {Friction} {Behaviour}.}, year = {1995} }
@inproceedings{faucris.223714651, author = {Geiger, Manfred and Meßner, Arthur and Engel, Ulf and Kals, Roland and Vollertsen, Frank}, booktitle = {Proceedings of the 9th International Cold Forging
Congress}, editor = {Starding, P.}, faupublication = {yes}, note = {LFT Import::2019-08-05 (550)}, pages = {155-164}, peerreviewed = {unknown}, publisher = {FMJ International Publications Ltd}, title
= {{Design} of {Micro}-{Forming} {Processes} - {Fundamentals}, {Material} {Data} and {Friction} {Behaviour}}, venue = {Solihull, UK}, year = {1995} } @inproceedings{faucris.235433395, author =
{Rathmann, Wigand and Kallweit, Micheal}, booktitle = {Presentations at the MatRIC Conference "Making and Communicating Mathematical Meaning"}, date = {2019-10-14/2019-10-15}, faupublication = {yes},
peerreviewed = {No}, title = {{Developing} a network for digital assessment}, url = {https://ucarecdn.com/2934c653-ed47-424d-9b2d-705395ec79d6/
DevelopinganetworkfordigitalcontentandassessmentWigandRathmann.pdf}, venue = {Bergen}, year = {2019} } @inproceedings{faucris.114040784, author = {Rathmann, Wigand}, booktitle = {7. SAXON SIMULATION
MEETING and Mathcad-Workshop}, date = {2015-03-31}, faupublication = {yes}, peerreviewed = {No}, publisher = {TU Chemnitz, Professur für Montage- und Handhabungstechnik}, title = {{Die} {Kettenlinie}
als {Kerbgeometrie}}, url = {http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-172556}, venue = {TU Chemnitz}, year = {2015} } @article{faucris.124012064, author = {Jahn, Johannes}, doi = {10.11650/
tjm.18.2014.4940}, faupublication = {yes}, journal = {Taiwanese Journal of Mathematics}, note = {UnivIS-Import:2015-04-02:Pub.2014.nat.dma.pama21.direct}, pages = {737 - 757}, peerreviewed = {Yes},
title = {{Directional} derivative in set optimization with the set less order relation}, volume = {19}, year = {2015} } @article{faucris.243564769, abstract = {Wearable sensor technology already has
a great impact on the endurance running community. Smartwatches and heart rate monitors are heavily used to evaluate runners’ performance and monitor their training progress. Additionally,
foot-mounted inertial measurement units (IMUs) have drawn the attention of sport scientists due to the possibility to monitor biomechanically relevant spatio-temporal parameters outside the lab in
real-world environments. Researchers developed and investigated algorithms to extract various features using IMU data of different sensor positions on the foot. In this work, we evaluate whether the
sensor position of IMUs mounted to running shoes has an impact on the accuracy of different spatio-temporal parameters. We compare both the raw data of the IMUs at different sensor positions as well
as the accuracy of six endurance running-related parameters. We contribute a study with 29 subjects wearing running shoes equipped with four IMUs on both the left and the right shoes and a motion
capture system as ground truth. The results show that the IMUs measure different raw data depending on their position on the foot and that the accuracy of the spatio-temporal parameters depends on
the sensor position. We recommend to integrate IMU sensors in a cavity in the sole of a running shoe under the foot’s arch, because the raw data of this sensor position is best suitable for the
reconstruction of the foot trajectory during a stride.}, author = {Zrenner, Markus and Küderle, Arne and Roth, Nils and Jensen, Ulf and Dümler, Burkhard and Eskofier, Björn}, doi = {10.3390/
s20195705}, faupublication = {yes}, journal = {Sensors}, keywords = {wearable computing; foot kinematics; sensor position; zero velocity update; inertial measurement unit; sport science; running},
peerreviewed = {Yes}, title = {{Does} the {Position} of {Foot}-{Mounted} {IMU} {Sensors} {Influence} the {Accuracy} of {Spatio}-{Temporal} {Parameters} in {Endurance} {Running}?}, volume = {20}, year
= {2020} } @incollection{faucris.117900684, abstract = {We consider some elementary model problems that are taken to be representative of more important models on complex spatial structures. We
discuss domain decomposition techniques from the point of view of optimal control in that coupling conditions are viewed as controllability constraints. This leads to the notion of virtual controls,
which has been introduced by J.L. Lions. We pursue an augmented Lagrangian point of view. By this method the iterative coupling turns into a sequence of PDE control problems. We also provide
extensions of the methods to elliptic problems on networked domains. This contribution is in honor of J.E. Lagnese, with whom the author collaborated over the past 15 years. Most of the results of
this paper have been obtained in this collaboration.}, address = {USA}, author = {Leugering, Günter}, booktitle = {Control Theory of Partial Differentiial Equations}, doi = {10.1201/
9781420028317.ch9}, editor = {Oleg Imanuvilov, Guenter Leugering, Roberto Triggiani, Bing-Yu Zhang}, faupublication = {yes}, isbn = {978-0-8247-2546-4}, month = {Jan}, pages = {125-155}, peerreviewed
= {unknown}, series = {A Series of Lecture Notes in Pure and Applied Mathematics}, title = {{Domain} decomposition in optimal control problems for partial differential equations revisited}, volume =
{242}, year = {2005} } @book{faucris.123698784, abstract = {We consider optimal control problems for elliptic systems under control constraints on networked domains. In particular, we study such
systems in a format that allows for applications in problems including membranes and Reissner-Mindlin plates on multi-link-domains, called networks. We first provide the models, derive first order
optimality conditions in terms of variational equations and inequalities for a control-constrained linear-quadratic optimal control problem, and then introduce a non-overlapping iterative domain
decomposition method, which is based on Robin-type interface updates at multiple joints (edges). We prove convergence of the iteration and derive a posteriori error estimates with respect to the
iteration across the interfaces.}, author = {Leugering, Günter}, doi = {10.1007/978-3-540-75199-1{\_}10}, faupublication = {yes}, isbn = {9783540751984}, pages = {119-130}, peerreviewed = {unknown},
title = {{Domain} decomposition of constrained optimal control problems for 2d elliptic system on networked domains: convergence and a posteriori error estimates}, volume = {60}, year = {2008} }
@article{faucris.117901784, abstract = {We consider optimal control problems related to exact- and approximate controllability of dynamic networks of elastic strings. In this note we concentrate on
problems with linear dynamics, no state and no control constraints. The emphasis is on approximating target states and velocities in part of the network using a dynamic domain decomposition method (d
(3)m) for the optimality system on the network. The decomposition is established via a Uzawa-type saddle-point iteration associated with an augmented Lagrangian relaxation of the transmission
conditions at multiple joints. We consider various cost functions and prove convergence of the infinite dimensional scheme for an exemplaric choice of the cost. We also give numerical evidence in the
case of simple exemplaric networks.}, author = {Leugering, Günter}, doi = {10.1023/A:1008721402512}, faupublication = {yes}, journal = {Computational Optimization and Applications}, keywords =
{dynamic networks of elastic strings;dynamic domain decomposition;optimal control}, pages = {5-27}, peerreviewed = {Yes}, title = {{Domain} decomposition of optimal control problems for dynamic
networks of elastic strings}, url = {http://link.springer.com/article/10.1023/A:1008721402512}, volume = {16}, year = {2000} } @incollection{faucris.118835024, address = {Ilmenau}, author = {Jahn,
Johannes}, booktitle = {27. Internationales Wissenschaftliches Kolloquium 1982}, editor = {G. Linnemann}, faupublication = {no}, note = {UnivIS-Import:2015-04-17:Pub.1982.nat.dma.pama21.dualit},
pages = {7-9}, peerreviewed = {unknown}, publisher = {Technische Hochschule Ilmenau}, title = {{Dualität} in der {Vektoroptimierung} mit {Anwendung} in der vektoriellen {Approximation}}, volume =
{5}, year = {1982} } @incollection{faucris.118842284, address = {Berlin}, author = {Jahn, Johannes}, booktitle = {Recent Advances and Historical Development of Vector Optimization}, editor = {J.
Jahn, W. Krabs}, faupublication = {no}, note = {UnivIS-Import:2015-04-17:Pub.1987.nat.dma.pama21.dualit}, pages = {160-172}, peerreviewed = {unknown}, publisher = {Springer}, series = {Lecture Notes
in Economics and Mathematical Systems}, title = {{Duality} in {Partially} {Ordered} {Sets}}, year = {1987} } @article{faucris.115105144, abstract = {In this paper the problem dual to a convex vector
optimization problem is defined. Under suitable assumptions, a weak, strong and strict converse duality theorem are proved. In the case of linear mappings the formulation of the dual is refined such
that well-known dual problems of Gale, Kuhn and Tucker [8] and Isermann [12] are generalized by this approach. © 1983 The Mathematical Programming Society, Inc.}, author = {Jahn, Johannes}, doi =
{10.1007/BF02594784}, faupublication = {no}, journal = {Mathematical Programming}, keywords = {Duality; Vector Optimization}, note = {UnivIS-Import:2015-03-05:Pub.1983.nat.dma.pama21.dualit}, pages =
{343-353}, peerreviewed = {Yes}, title = {{Duality} in {Vector} {Optimization}}, volume = {25}, year = {1983} } @article{faucris.214606914, abstract = {Consider a star-shaped network of strings. Each
string is governed by the wave equation. At each boundary node of the network there is a player that performs Dirichlet boundary control action and in this way influences the system state. At the
central node, the states are coupled by algebraic conditions in such a way that the energy is conserved. We consider the corresponding antagonistic game where each player minimizes a certain
quadratic objective function that is given by the sum of a control cost and a tracking term for the final state. We prove that under suitable assumptions a unique Nash equilibrium exists and give an
explicit representation of the equilibrium strategies.}, author = {Gugat, Martin and Steffensen, Sonja}, doi = {10.1051/cocv/2017082}, faupublication = {yes}, journal = {Esaim-Control Optimisation
and Calculus of Variations}, note = {CRIS-Team WoS Importer:2019-03-26}, pages = {1789-1813}, peerreviewed = {Yes}, title = {{DYNAMIC} {BOUNDARY} {CONTROL} {GAMES} {WITH} {NETWORKS} {OF} {STRINGS}},
volume = {24}, year = {2018} } @article{faucris.117902004, abstract = {This paper is concerned with dynamic domain decomposition for optimal boundary control and for approximate and exact boundary
controllability of wave propagation in heterogeneous media. We consider a cost functional which penalizes the deviation of the final state of the solution of the global problem from a specified
target state. For any fixed value of the penalty parameter, optimality conditions are derived for both the global optimal control problem and for local optimal control problems obtained by a domain
decomposition and a saddle-point-type iteration. Convergence of the iterations to the solution of the global optimality system is established. We then pass to the limit in the iterations as the
penalty parameter increases without bound and show that the limiting local iterations converge to the solution of the optimality system associated with the problem of finding the minimum norm control
that drives the solution of the global problem to a specified target state.}, author = {Langnese, John E. and Leugering, Günter}, doi = {10.1137/S0363012998333530}, faupublication = {no}, journal =
{SIAM Journal on Control and Optimization}, keywords = {domain decomposition;optimal control;controllability;saddle-point iteration; 93B05; 93B40; 49N10; 65K10}, pages = {503-537}, peerreviewed =
{Yes}, title = {{Dynamic} domain decomposition in approximate and exact boundary control in problems of transmission for wave equations}, volume = {38}, year = {2000} } @article{faucris.117903324,
abstract = {We consider general networks of strings and/or Timoshenko beams. We apply controls at boundary nodes of the network and want to minimize some cost function along (part of) the structure.
Optimality systems for the entire structure are far too complex to compute in reasonable time. In particular, in real-time applications one wants to reduce the size of the problem. Thus, dynamic
decomposition into its physical elements appears to be a natural approach. We show how to iteratively decompose the global optimality system into a system related to a substructure. Then we interpret
the local system as an optimality system corresponding to an optimal control problem for the substructure and finally we show convergence of the "outer" iteration.}, author = {Leugering, Günter}, doi
= {10.1137/S0363012997331986}, faupublication = {no}, journal = {SIAM Journal on Control and Optimization}, keywords = {dynamic domain decomposition;networks of strings and beams;saddle-point
iteration;relaxation}, month = {Jan}, pages = {1649-1675}, peerreviewed = {Yes}, title = {{Dynamic} domain decomposition of optimal control problems for networks of strings and {Timoshenko} beams},
url = {http://epubs.siam.org/doi/abs/10.1137/S0363012997331986}, volume = {37}, year = {1999} } @inproceedings{faucris.214864375, abstract = {In e-Learning environments, the Learning Management
Systems (LMS) play a central role.
The LMS Moodle and ILIAS are widely used at universities and Universities of Applied
Sciences. But even nowadays, it is still a challenge to use the possibilities given by digital
materials to create learning content which offers more than a classical textbook. Several
tools are available to show, explain and dynamically explore mathematical concepts,
among them are CoCalc (former SageMathCloud), SageCell, CDF-Player, JSXGraph and
GeoGebra. It is still quite common that content for a lecture is provided inside the local
LMS, while the learner has to open another platform or software to use the dynamic
mathematical tools for exploration of the content. In our contribution (talk or poster) we
will show our approaches to include dynamic mathematics inside LMS.
In particular, we use JSXGraph for dynamic 2D diagrams and the SageMathCell for a wide
range of mathematical aspects. JSXGraph is a JavaScript library, which is quite easy to use
and which offers the possibility to include user interactions via HTML-forms. The
advantage is that the computations are done in the browser on client-side. On the other
hand, SageMathCell has a huge variety of tools already included, like Maxima, R, Octave
or Python. Furthermore, it is very easy to show 3D-diagrams. This is made possible by
executing the computation on a remote server, which has to be connected from the content.
We include JSXGraph and SageMathCell in classical HTML-Pages as media content and
through dedicated plug-ins for the LMS Moodle and ILIAS. During the ”Mathematics for
Engineers” (part 1-3) we provide additional material for several topics, e.g. series, (un)-
constrained optimization, integration, parametrization of curves and surfaces or differential
equations. We see the advantage that the diagrams will become an integral part of the
learning modules and the learning unit does not look like patchwork.
Further, the learners can focus on the content and do not have to bring additional software
or to login in a second learning platform. During presence lectures the lecturer can use the
same LMS as the learnersdo for their follow-up work.
For JSXGraph, there is a plug-in for Moodle, for the usage of SageMathCell inside ILIAS
the first author could initiate the development.
An integrodifferential equation of the Volterra type is considered under the action of anL[2](0, T, L[2](Γ))-boundary control. By harmonic analysis arguments it is shown that the controllability
results obtained in [17] for the underlying reference model associated with a trivial convolution kernel, carry over to the model under consideration without any smallness assumption concerning the
memory kernel.
}, author = {Leugering, Günter}, doi = {10.1007/BF01442653}, faupublication = {no}, journal = {Applied Mathematics and Optimization}, pages = {223-250}, peerreviewed = {Yes}, title = {{EXACT}
{BOUNDARY} {CONTROLLABILITY} {OF} {AN} {INTEGRODIFFERENTIAL} {EQUATION}}, url = {http://link.springer.com/article/10.1007/BF01442653}, volume = {15}, year = {1987} } @article{faucris.227770910,
abstract = {The exact boundary controllability for hyperbolic systems can not be realized generally on a network with loops (see [16]). In this paper we consider the exact boundary controllability of
nodal profile on a network with loops. Precisely speaking, on a network with a triangle-like loop, when nodal profiles are given at various kinds of nodes, different constructive methods can be used
to get the corresponding exact boundary controllability of nodal profile for Saint-Venant system by means of boundary controls acting on suitable nodes, respectively. This reveals that the exact
boundary controllability of nodal profile is quite different from the usual exact boundary controllability, and has relatively distinctive behaviors and characters. (C) 2018 Elsevier Masson SAS. All
rights reserved.}, author = {Zhuang, Kaili and Leugering, Günter and Li, Tatsien}, doi = {10.1016/j.matpur.2018.10.001}, faupublication = {yes}, journal = {Journal De Mathematiques Pures Et
Appliquees}, note = {CRIS-Team WoS Importer:2019-10-11}, pages = {34-60}, peerreviewed = {Yes}, title = {{Exact} boundary controllability of nodal profile for {Saint}-{Venant} system on a network
with loops}, volume = {129}, year = {2019} } @article{faucris.262156508, abstract = {Based on the theory of exact boundary controllability of nodal profile for hyperbolic systems, the authors propose
the concept of exact boundary controllability of partial nodal profile to expand the scope of applications. With the new concept, we can shorten the controllability time, save the number of controls,
and increase the number of charged nodes with given nodal profiles. Furthermore, we introduce the concept of in-situ controlled node to deal with a new situation that one node can be charged and
controlled simultaneously. The minimum number of boundary controls on the entire tree-like network is determined by using the concept of ‘degree of freedom of charged nodes’ introduced. And the
concept of ‘control path’ is introduced to appropriately divide the network, so that we can determine the infimum of controllability time. General frameworks of constructive proof are given on a
single interval, a star-like network, a chain-like network and a planar tree-like network for linear wave equation(s) with Dirichlet, Neumann, Robin and dissipative boundary conditions to establish a
complete theory on the exact boundary controllability of partial nodal profile.}, author = {Wang, Yue and Li, Tatsien}, doi = {10.1016/j.nonrwa.2021.103383}, faupublication = {yes}, journal =
{Nonlinear Analysis-Real World Applications}, keywords = {Control path; Exact boundary controllability; Partial nodal profile; Piecewise C; Planar tree-like network; The degree of freedom of charged
node}, note = {CRIS-Team Scopus Importer:2021-07-30}, peerreviewed = {Yes}, title = {{Exact} boundary controllability of partial nodal profile for network of strings}, volume = {62}, year = {2021} }
@article{faucris.119702924, abstract = {This paper concerns a system of equations describing the vibrations of a planar network of nonlinear Timoshenko beams. The authors derive the equations and
appropriate nodal conditions, determine equilibrium solutions and, using the methods of quasilinear hyperbolic systems, prove that for tree-like networks the natural initial-boundary value problem
admits semi-global classical solutions in the sense of Li [Li, T. T., Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Ser. Appl. Math., vol 3, American Institute of
Mathematical Sciences and Higher Education Press, 2010] existing in a neighborhood of the equilibrium solution. The authors then prove the local exact controllability of such networks near such
equilibrium configurations in a certain specified time interval depending on the speed of propagation in the individual beams.}, author = {Leugering, Günter and Li, Tatsien and Gu, Qilong}, doi =
{10.1007/s11401-017-1092-7}, faupublication = {yes}, journal = {Chinese Annals of Mathematics. Series B}, keywords = {Exact boundary controllability; Nonlinear Timoshenko beams; Semi-global classical
solutions; Tree-like networks}, pages = {711-740}, peerreviewed = {unknown}, title = {{Exact} boundary controllability on a tree-like network of nonlinear planar {Timoshenko} beams}, volume = {38},
year = {2017} } @article{faucris.117905084, abstract = {Distributed load control of motions governed by an abstract partial integrodifferential equation is studied. First a new system-theoretic
approach to a mild solution theory of equations arising in viscoelasticity of fading memory type is developed. Then it is shown that control results for the damped abstract wave equation devolve to a
special class of “virgin” material with fading memory.}, author = {Leugering, Günter}, doi = {10.1080/00036818408839521}, faupublication = {no}, journal = {Applicable Analysis}, keywords = {93B05;
45K05; 73K05}, month = {Jan}, pages = {221-243}, peerreviewed = {Yes}, title = {{EXACT} {CONTROLLABILITY} {IN} {VISCOELASTICITY} {OF} {FADING} {MEMORY} {TYPE}}, url = {http://www.tandfonline.com/doi/
abs/10.1080/00036818408839521}, volume = {18}, year = {1984} } @article{faucris.121753104, abstract = {We study optimal control problems for linear systems with prescribed initial and terminal
states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm-minimal exact control can be computed as the solution of an
optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the
application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley & Sons,
Ltd.}, author = {Gugat, Martin and Zuazua, Enrique}, doi = {10.1002/oca.2238}, faupublication = {yes}, journal = {Optimal Control Applications & Methods}, keywords = {exact controllability;optimal
control;terminal constraint;exact penalization;wave equation;heat equation;moment equations;method of moments;L1 optimal control;abstract Cauchy problems;nonsmooth optimization}, peerreviewed =
{Yes}, title = {{Exact} penalization of terminal constraints for optimal control problems}, url = {http://onlinelibrary.wiley.com/doi/10.1002/oca.2238/abstract}, volume = {37}, year = {2016} }
@inproceedings{faucris.263538379, abstract = {In this paper, we study the time-fractional diffusion equation on a metric star graph. The existence and uniqueness of the weak solution are investigated
and the proof is based on eigenfunction expansions. Some priori estimates and regularity results of the solution are proved.}, author = {Mehandiratta, Vaibhav and Mehra, Mani and Leugering, Günter},
booktitle = {Communications in Computer and Information Science}, date = {2020-09-10/2020-09-12}, doi = {10.1007/978-981-16-4772-7{\_}2}, editor = {Ashish Awasthi, Sunil Jacob John, Satyananda
Panda}, faupublication = {yes}, isbn = {9789811647710}, keywords = {Caputo fractional derivative; Star graph; Time-fractional diffusion equation; Weak solution}, note = {CRIS-Team Scopus
Importer:2021-09-03}, pages = {25-41}, peerreviewed = {unknown}, publisher = {Springer Science and Business Media Deutschland GmbH}, title = {{Existence} and {Uniqueness} of {Time}-{Fractional}
{Diffusion} {Equation} on a {Metric} {Star} {Graph}}, venue = {Virtual, Online}, volume = {1345}, year = {2021} } @article{faucris.217952993, abstract = {In this paper, we study a nonlinear Caputo
fractional boundary value problem on a star graph. By means of transformation, an equivalent system of fractional boundary value problem is obtained. Then we establish the existence and uniqueness
results by fixed point theory: in particular, using Banach's contraction principle and Schaefer's fixed point theorem. We also present some examples to illustrate the application of our results.},
author = {Mehandiratta, Vaibhav and Mehra, Mani and Leugering, Günter}, doi = {10.1016/j.jmaa.2019.05.011}, faupublication = {yes}, journal = {Journal of Mathematical Analysis and Applications},
keywords = {Boundary value problem; Caputo fractional derivative; Differential equation on graphs; Fixed point theorems}, note = {CRIS-Team Scopus Importer:2019-05-21}, peerreviewed = {Yes}, title =
{{Existence} and uniqueness results for a nonlinear {Caputo} fractional boundary value problem on a star graph}, year = {2019} } @article{faucris.122946824, abstract = {We consider the flow of gas
through pipelines controlled by a compressor station. Under a subsonic flow assumption we prove the existence of classical solutions for a given finite time interval. The existence result is used to
construct Riemannian feedback laws and to prove a stabilization result for a coupled system of gas pipes with a compressor station. We introduce a Lyapunov function and prove exponential decay with
respect to the L^2-norm.}, author = {Gugat, Martin and Herty, Michael}, doi = {10.1051/cocv/2009035}, faupublication = {yes}, journal = {Esaim-Control Optimisation and Calculus of Variations},
keywords = {Classical solution; networked hyperbolic systems; gas networks; feedback law; Lyapunov function;}, pages = {28-51}, peerreviewed = {Yes}, title = {{Existence} of classical solutions and
feedback stabilization for the flow in gas networks}, url = {http://www.esaim-cocv.org/index.php?option=article&access=standard&Itemid=129&url=/articles/cocv/abs/first/cocv0926/cocv0926.html}, volume
= {17}, year = {2011} } @inproceedings{faucris.106690584, author = {Hante, Falk and Sigalotti, Mario}, booktitle = {Decision and Control (CDC) 2010. CDC 2010, 49th IEEE Conference on}, faupublication
= {no}, pages = {5668--5673}, peerreviewed = {unknown}, title = {{Existence} of common {Lyapunov} functions for infinite-dimensional switched linear systems}, year = {2010} } @article
{faucris.257707422, abstract = {In this paper, we study a nonlinear fractional boundary value problem on a particular metric graph, namely, a circular ring with an attached edge. First, we prove
existence and uniqueness of solutions using the Banach contraction principle and Krasnoselskii's fixed point theorem. Further, we investigate different kinds of Ulam-type stability for the proposed
problem. Finally, an example is given in order to demonstrate the application of the obtained theoretical results.}, author = {Mehandiratta, Vaibhav and Mehra, Mani and Leugering, Günter}, doi =
{10.3934/nhm.2021003}, faupublication = {yes}, journal = {Networks and Heterogeneous Media}, note = {CRIS-Team WoS Importer:2021-05-07}, pages = {155-185}, peerreviewed = {Yes}, title = {{EXISTENCE}
{RESULTS} {AND} {STABILITY} {ANALYSIS} {FOR} {A} {NONLINEAR} {FRACTIONAL} {BOUNDARY} {VALUE} {PROBLEM} {ON} {A} {CIRCULAR} {RING} {WITH} {AN} {ATTACHED} {EDGE} : {A} {STUDY} {OF} {FRACTIONAL}
{CALCULUS} {ON} {METRIC} {GRAPH}}, volume = {16}, year = {2021} } @article{faucris.110994884, author = {Jahn, Johannes and Khan, Akhtar Ali}, faupublication = {yes}, journal = {Journal of Nonlinear
and Convex Analysis}, note = {UnivIS-Import:2015-03-09:Pub.2002.nat.dma.pama21.existe}, pages = {315-330}, peerreviewed = {Yes}, title = {{Existence} {Theorems} and {Characterizations} of
{Generalized} {Contingent} {Epiderivatives}}, volume = {3}, year = {2002} } @article{faucris.115113284, abstract = {In this paper, existence theorems for minimal, weakly minimal, and properly minimal
elements of a subset of a partially-ordered, real linear space are presented. © 1986 Plenum Publishing Corporation.}, author = {Jahn, Johannes}, doi = {10.1007/BF00938627}, faupublication = {no},
journal = {Journal of Optimization Theory and Applications}, keywords = {approximation; scalarization; Vector optimization}, note = {UnivIS-Import:2015-03-05:Pub.1986.nat.dma.pama21.existe}, pages =
{397-406}, peerreviewed = {Yes}, title = {{Existence} {Theorems} in {Vector} {Optimization}}, volume = {50}, year = {1986} } @article{faucris.202369309, author = {Keimer, Alexander and Pflug, Lukas
and Spinola, Michele}, doi = {10.1016/j.jmaa.2018.05.013}, faupublication = {yes}, journal = {Journal of Mathematical Analysis and Applications}, keywords = {Method of characteristics; Fixed-point
problem; Nonlocal conservation law; Multi-dimensional nonlocal balance law; Well-posedness; Regularity}, pages = {18-55}, peerreviewed = {Yes}, title = {{Existence}, uniqueness and regularity of
multi-dimensional nonlocal balance laws with damping}, volume = {466}, year = {2018} } @article{faucris.202369670, author = {Keimer, Alexander and Pflug, Lukas}, doi = {10.1016/j.jde.2017.05.015},
faupublication = {yes}, journal = {Journal of Differential Equations}, keywords = {Nonlocal balance law; Existence; Regularity; Nonlocal conservation law; Weak solution; Uniqueness}, pages =
{4023-4069}, peerreviewed = {Yes}, title = {{Existence}, uniqueness and regularity results on nonlocal balance laws}, volume = {263}, year = {2017} } @incollection{faucris.114813204, address =
{Ilmenau}, author = {Jahn, Johannes}, booktitle = {30. Internationales Wissenschaftliches Kolloquium 1985}, editor = {G. Linnemann}, faupublication = {no}, note =
{UnivIS-Import:2015-04-17:Pub.1985.nat.dma.pama21.existe}, pages = {67-69}, peerreviewed = {unknown}, publisher = {Technische Hochschule Ilmenau}, title = {{Existenzsätze} in der
{Vektoroptimierung}}, year = {1985} } @article{faucris.120845164, author = {Hante, Falk and Amin, Saurabh and Bayen, Alexandre M.}, doi = {10.1109/TAC.2011.2158171}, faupublication = {no}, journal =
{IEEETransactions on Automatic Control}, pages = {291--301}, peerreviewed = {Yes}, title = {{Exponential} stability of switched linear hyperbolic initial-boundary value problems}, volume = {57}, year
= {2012} } @article{faucris.117724244, abstract = {We consider the problem of boundary feedback stabilization of a vibrating string that is fixed at one end and with control action at the other end.
In contrast to previous studies that have required L^2-regularity for the initial position and H^-1-regularity for the initial velocity, in this paper we allow for initial positions with L^
1-regularity and initial velocities in W^-1, 1 on the space interval. It is well known that for a certain feedback parameter, for sufficiently regular initial states the classical energy of the
closed-loop system with Neumann velocity feedback is controlled to zero after a finite time that is equal to the minimal time where exact controllability holds. In this paper, we present a Dirichlet
boundary feedback that yields a well-defined closed-loop system in the (L^1, W^-1,1) framework and also has this property. Moreover, for all positive feedback parameters our feedback law leads to
exponential decay of a suitably defined L^1-energy. For more regular initial states with (L^2, H^-1) regularity, the proposed feedback law leads to exponential decay of an energy that corresponds to
this framework. If the initial states are even more regular with H^1-regularity of the initial position and L^2-regularity of the initial velocity, our feedback law also leads to exponential decay of
the classical energy.}, author = {Gugat, Martin}, doi = {10.1137/140977023}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, keywords = {Dirichlet boundary control;
Energy decay; Exact control; Exponential stability; Vibrating string}, note = {UnivIS-Import:2015-03-11:Pub.2015.nat.dma.zentr.expone}, pages = {526-546}, peerreviewed = {Yes}, title = {{EXPONENTIAL}
{STABILIZATION} {OF} {THE} {WAVE} {EQUATION} {BY} {DIRICHLET} {INTEGRAL} {FEEDBACK}}, volume = {53}, year = {2015} } @inproceedings{faucris.118501504, abstract = {We consider the feedback
stabilization of quasilinear hyperbolic systems on star-shaped networks. We present boundary feedback controls with varying delays. The delays are given by C^1-functions with bounded derivatives. We
obtain the existence of unique C^1-solutions on a given finite time interval. In order to measure the system evolution, we introduce an L^2-Lyapunov function with delay terms. The feedback controls
yield the exponential decay of the Lyapunov function with time. This implies the exponential stability of the system. Our results can be applied on the stabilization of the isothermal Euler equations
with friction that model the gas flow in pipe networks. © 2012 IEEE.}, author = {Hirsch-Dick, Markus and Gugat, Martin and Leugering, Günter}, booktitle = {Methods and Models in Automation and
Robotics (MMAR)}, date = {2012-08-27/2012-08-30}, doi = {10.1109/MMAR.2012.6347931}, faupublication = {yes}, keywords = {Couplings; Delay; Feedback control; Lyapunov methods; Mathematical model;
Propagation;}, note = {UnivIS-Import:2015-04-16:Pub.2012.nat.dma.lama1.feedba}, pages = {125-130}, peerreviewed = {Yes}, series = {IEEEXplore}, title = {{Feedback} stabilization of quasilinear
hyperbolic systems with varying delays}, url = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6347931&isnumber=6347808}, venue = {Miedzyzdroje, Poland}, year = {2012} } @article
{faucris.118518884, abstract = {This work presents the application of a Fully Implicit Method for Ostwald Ripening (FIMOR) for simulating the ripening of ZnO quantum dots (QDs). Its stable numerics
allow FIMOR to employ the full exponential term of the Gibbs–Thomson equation which significantly outperforms the common Taylor-approximation at typical QD sizes below 10 nm. The implementation is
consistent with experimental data for temperatures between 10 and 50 °C and the computational effort is reduced by a factor of 100–1000 compared to previous approaches. This reduced the simulation
time on a standard PC from several hours to a few minutes. In the second part, we demonstrate the high potential and accuracy of FIMOR by its application to several challenging studies. First, we
compare numeric results obtained for ripening of ZnO QDs exposed to temperature ramps with experimental data. The deviation between simulation and experiment in the mean volume weighted particle size
was as small as 5\%. Second, a map for the process parameter space spanned by ripening time and temperature is created based on a large number (>50) of FIMOR runs. From this map appropriate process
parameters to adjust a desired dispersity are easily deduced. Further data analysis reveals in agreement with literature findings that the particle size distribution converges towards a
self-preserving stable shape. Equations describing the time dependent particle size distribution with high accuracy are presented. Finally, we realized the transfer from low volume batch experiments
to continuous QD processing. We modeled the continuous ZnO synthesis in a fully automated microreaction plant and found an excellent agreement between the numeric prediction and the experimental
results by considering the residence time distribution.}, author = {Haderlein, Michael and Segets, Doris and Gröschel, Michael and Pflug, Lukas and Leugering, Günter and Peukert, Wolfgang}, doi =
{10.1016/j.cej.2014.09.040}, faupublication = {yes}, journal = {Chemical Engineering Journal}, pages = {706--715}, peerreviewed = {Yes}, title = {{FIMOR}: {An} efficient simulation for {ZnO} quantum
dot ripening applied to the optimization of nanoparticle synthesis}, volume = {260}, year = {2015} } @article{faucris.123271104, abstract = {It appears that most models for micro-structured materials
with auxetic deformations were found by clever intuition, possibly combined with optimization tools, rather than by systematic searches of existing structure archives. Here we review our recent
approach of finding micro-structured materials with auxetic mechanisms within the vast repositories of planar tessellations. This approach has produced two previously unknown auxetic mechanisms,
which have Poisson's ratio v(ss) = -1 when realized as a skeletal structure of stiff incompressible struts pivoting freely at common vertices. One of these, baptized Triangle-Square Wheels, has been
produced as a linear-elastic cellular structure from Ti-6Al-4V alloy by selective electron beam melting. Its linear-elastic properties were measured by tensile experiments and yield an effective
Poisson's ratio v(LE) approximate to -0.75, also in agreement with finite element modeling. The similarity between the Poisson's ratios v(SS) of the skeletal structure and v(LE) of the linear-elastic
cellular structure emphasizes the fundamental role of geometry for deformation behavior, regardless of the mechanical details of the system. The approach of exploiting structure archives as candidate
geometries for auxetic materials also applies to spatial networks and tessellations and can aid the quest for inherently three-dimensional auxetic mechanisms.}, author = {Mitschke, Holger and
Schwerdtfeger, Jan and Schury, Fabian and Stingl, Michael and Körner, Carolin and Singer, Robert and Robins, Vanessa and Mecke, Klaus and Schröder-Turk, Gerd}, doi = {10.1002/adma.201100268},
faupublication = {yes}, journal = {Advanced Materials}, pages = {2669-2674}, peerreviewed = {Yes}, title = {{Finding} {Auxetic} {Frameworks} in {Periodic} {Tessellations}}, volume = {23}, year =
{2011} } @inproceedings{faucris.122718904, abstract = {
For master students of the process technology and energy technology we offer a course on simulation of transport processes. Two aims are focused:
The students should learn or repeat the basics in using a simulation tools.
They should learn to combine specialized knowledge with mathematical concepts to simulate a technical process.
In the first part of the course the students collect first experiences in programming by modeling a Rankine cycle using Matlab. Hereby they discuss the quality of the presented mathematical model and
possible improvements. The students are accompanied by the lecturers of the lecture "Transport processes" (master studies) and “Mathematics for Engineers” (bachelor studies). While working on the
computer learners can discuss their questions, programming ideas and difficulties with each other and the trainers and they receive immediate feedbacks. The second part of the course is focuses on
simple heat transfer processes described by ODE or PDE models. Therefore some mathematical concepts (e.g. linear equation systems) are needed, but they have learned it about three years ago.
Up to now the mathematical models of the technical processes and the repetition of the mathematical basics were presented as talks during the lectures with little time remained for the discussion.
But the students need enough time to reconsider the introduced topics. We have seen, that this process needs more time then scheduled, e.g. to understand the way to describe the first derivative of a
function on a interval by a system of linear equation system.
Now we are working on a didactic reorganization (flipped classroom) of our course, so that the students will have enough time, to repeat subjects needed in our course.
Therefore on-line learning modules and formative on-line tests have to be finished to prepare the single sessions. Thus questions can be discussed quite early and it remains more time to discuss,
e.g. improvements of the models. By using the flipped classroom concept the students will be motivated to work more independently and they may reach a better understanding by using the digital
learning materials as starting point for the considerations. Thus we will generate a more active participation in the teaching and learning events and better learner’s results can be achieved.
In my talk I would like to discuss these ideas and I will give a short status report of the actual course.
}, author = {Rathmann, Wigand and Wensing, Michael and Zepf, Stefanie}, booktitle = {The 18th SEFI Mathematics working Group Seminar}, date = {2016-06-27/2016-06-29}, faupublication = {yes},
peerreviewed = {unknown}, title = {{Flipped} classroom in interdisciplinary course}, url = {http://www.math.chalmers.se/SEFIMWG2016/Book%20of%20accepted%20abstracts/part49.htm}, venue = {Gothenburg},
year = {2016} } @article{faucris.117348704, abstract = {We consider a network of pipelines where the flow is controlled by a number of compressors. The consumer demand is described by desired
boundary traces of the system state. We present conditions that guarantee the existence of compressor controls such that after a certain finite time the state at the consumer nodes is equal to the
prescribed data. We consider this problem in the framework of continuously differentiable states. We give an explicit construction of the control functions for the control of compressor stations in
gas distribution networks. Copyright © 2010 John Wiley & Sons, Ltd.}, author = {Gugat, Martin and Herty, Michael and Schleper, Veronika}, doi = {10.1002/mma.1394}, faupublication = {yes}, journal =
{Mathematical Methods in the Applied Sciences}, keywords = {exact controllability;nodal control of networked hyperbolic systems;nodal profile control;classical solutions}, note =
{UnivIS-Import:2015-03-09:Pub.2010.nat.dma.zentr.flowco}, pages = {745-757}, peerreviewed = {Yes}, title = {{Flow} control in gas networks: {Exact} controllability to a given demand}, url = {http://
onlinelibrary.wiley.com/doi/10.1002/mma.1394/abstract}, volume = {34}, year = {2010} } @incollection{faucris.119059864, address = {Heidelberg}, author = {Eichfelder, Gabriele and Jahn, Johannes},
booktitle = {Nonlinear Analysis and Variational Problems}, editor = {P.M. Pardalos, T.M. Rassias, A.A. Khan}, faupublication = {yes}, isbn = {978-1-4419-0157-6}, note =
{UnivIS-Import:2015-04-20:Pub.2009.nat.dma.zentr.founda}, pages = {259-284}, peerreviewed = {No}, publisher = {Springer}, title = {{Foundations} of {Set}-{Semidefinite} {Optimization}}, url = {http:/
/www.springer.com/mathematics/applications/book/978-1-4419-0157-6}, year = {2010} } @article{faucris.247573907, abstract = {In this paper, we study optimal control problems for nonlinear fractional
order boundary value problems on a star graph, where the fractional derivative is described in the Caputo sense. The adjoint state and the optimality system are derived for fractional optimal control
problem (FOCP) by using the Lagrange multiplier method. Then, the existence and uniqueness of solution of the adjoint equation is proved by means of the Banach contraction principle. We also present
a numerical method to find the approximate solution of the resulting optimality system. In the proposed method, the L2 scheme and the GrUnwald-Letnikov formula is used for the approximation of the
Caputo fractional derivative and the right Riemann-Liouville fractional derivative, respectively, which converts the optimality system into a system of linear algebraic equations. Two examples are
provided to demonstrate the feasibility of the numerical method.}, author = {Mehandiratta, Vaibhav and Mehra, Mani and Leugering, Günter}, doi = {10.3934/mcrf.2020033}, faupublication = {yes},
journal = {Mathematical Control and Related Fields}, note = {CRIS-Team WoS Importer:2021-01-08}, pages = {189-209}, peerreviewed = {Yes}, title = {{FRACTIONAL} {OPTIMAL} {CONTROL} {PROBLEMS} {ON} {A}
{STAR} {GRAPH}: {OPTIMALITY} {SYSTEM} {AND} {NUMERICAL} {SOLUTION}}, volume = {11}, year = {2021} } @inproceedings{faucris.108413624, author = {Rathmann, Wigand and Vogel, Frank and Ritter, Stefan},
booktitle = {19. CAD-FEM User's Meeting}, faupublication = {no}, peerreviewed = {No}, title = {{Free} {Form} {Shape} {Shell} {Optimization}}, year = {2001} } @inproceedings{faucris.117639104,
abstract = {In this article, we present the Free Material Optimization (FMO) problem for plates and shells based on Naghdi's shell model. In FMO - a branch of structural optimization - we search for
the ultimately best material properties in a given design domain loaded by a set of given forces. The optimization variable is the full material tensor at each point of the design domain. We give a
basic formulation of the problem and prove existence of an optimal solution. Lagrange duality theory allows to identify the basic problem as the dual of an infinite-dimensional convex nonlinear
semidefinite program. After discretization by the finite element method the latter problem can be solved using a nonlinear SDP code. The article is concluded by a few numerical studies.}, author =
{Gaile, Stefanie and Leugering, Günter and Stingl, Michael}, booktitle = {Proceedings of the 23rd IFIP TC 7 Conference on System Modelling and Optimization}, date = {2007-07-23/2007-07-27},
faupublication = {yes}, month = {Jan}, pages = {239-250}, peerreviewed = {unknown}, publisher = {Springer Verlag}, title = {{Free} material optimization for plates and shells}, venue = {Cracow},
volume = {312}, year = {2009} } @article{faucris.117640424, abstract = {Free material design deals with the question of finding the lightest structure subject to one or more given loads when both the
distribution of material and the material itself can be freely varied. We additionally consider constraints on local stresses in the optimal structure. We discuss the choice of formulation of the
problem and the stress constraints. The chosen formulation leads to a mathematical program with matrix inequality constraints, so-called nonlinear semidefinite program. We present an algorithm that
can solve these problems. The algorithm is based on a generalized augmented Lagrangian method. A number of numerical examples demonstrate the effect of stress constraints in free material
optimization.}, author = {Kocvara, Michal and Stingl, Michael}, doi = {10.1007/s00158-007-0095-5}, faupublication = {yes}, journal = {Structural and Multidisciplinary Optimization}, keywords =
{topology optimization;material optimization;stress based design;nonlinear semidefinite programming}, pages = {323-335}, peerreviewed = {Yes}, title = {{Free} material optimization for stress
constraints}, volume = {33}, year = {2007} } @article{faucris.124098084, abstract = {We present a compact overview of the recent development in free material optimization (FMO), a branch of
structural optimization. The goal of FMO is to design the ultimately best material (its mechanical properties and distribution in space) for a given purpose. We show that the current FMO models
naturally lead to linear and non-linear semidefinite programming problems (SDP); their numerical tractability is then guaranteed by recently introduced SDP algorithms.}, author = {Kocvara, Michal and
Stingl, Michael and Zowe, Jochem}, doi = {10.1080/02331930701778908}, faupublication = {yes}, journal = {Optimization}, keywords = {optimization of elastic structures;material optimization;topology
optimization;semidefinite programming;method of augmented Lagrangians}, pages = {79-100}, peerreviewed = {Yes}, title = {{Free} material optimization: recent progress}, volume = {57}, year = {2008} }
@article{faucris.107387104, abstract = {The goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure.
A natural formulation of this problem as a linear semidefinite program turns out to be numerically intractable. As an alternative, we propose a new approach, which is based on a nonlinear
semidefinite low-rank approximation of the semidefinite dual. We introduce an algorithm based on this approach and analyze its convergence properties. The article is concluded by numerical
experiments proving the effectiveness of the new approach}, author = {Stingl, Michael and Kocvara, Michal and Leugering, Günter}, doi = {10.1137/080717122}, faupublication = {yes}, journal = {SIAM
Journal on Optimization}, pages = {524--547}, peerreviewed = {Yes}, title = {{Free} {Material} {Optimization} with {Fundamental} {Eigenfrequency} {Constraints}}, volume = {20}, year = {2009} }
@inproceedings{faucris.239916510, author = {Peukert, Wolfgang and Segets, Doris and Haderlein, Michael and Pflug, Lukas and Leugering, Günter and Gröschel, Michael}, booktitle = {7th World Congress
on Particle Technology, WCPT 2014}, date = {2014-05-19/2014-05-22}, doi = {10.1016/j.proeng.2015.01.129}, faupublication = {yes}, keywords = {in situ; characterization; process design; optical
properties; quantum dots}, pages = {575-581}, peerreviewed = {No}, publisher = {Elsevier Ltd}, title = {{From} in situ characterization to process control of quantum dot systems}, venue = {Bejing},
volume = {102}, year = {2015} } @incollection{faucris.121892804, abstract = {Many important topics in multiobjective optimization and decision making have been studied in this book so far. In this
chapter, we wish to discuss some new trends and challenges which the field is facing. For brevity, we here concentrate on three main issues: new problem areas in which multiobjective optimization can
be of use, new procedures and algorithms to make efficient and useful applications of multiobjective optimization tools and, finally, new interesting and practically usable optimality concepts. Some
research has already been started and some such topics are also mentioned here to encourage further research. Some other topics are just ideas and deserve further attention in the near future. © 2008
Springer Berlin Heidelberg.}, address = {Berlin}, author = {Jahn, Johannes and Miettinen, K. and Ogryczak, W. and Deb, K. and Shimoyama, K. and Vetschera, R.}, booktitle = {Multiobjective
Optimization - Interactive and Evolutionary Approaches}, doi = {10.1007/978-3-540-88908-3-16}, editor = {J. Branke, K. Deb, K. Miettinen, R. Slowinski}, faupublication = {yes}, isbn = {3540889078},
note = {UnivIS-Import:2015-04-20:Pub.2008.nat.dma.pama21.future}, pages = {435-461}, peerreviewed = {unknown}, publisher = {Springer-verlag}, series = {Lecture Notes in Computer Science}, title =
{{Future} {Challenges}}, volume = {5252 LNCS}, year = {2008} } @article{faucris.119409444, abstract = {The problem of maximizing the performance in a fiber distributed data interface (FDDI) computer
network is formulated as a cooperative n-player game. Solutions of this game can be obtained by solving special optimization problems. Using the models and formulas developed by Tangemann (Ref. 1)
and Klehmet (Ref. 2) for the mean waiting times, the resulting optimization problems are presented and numerical results are given.}, author = {Jahn, Johannes and Häßler, S.}, faupublication = {yes},
journal = {Journal of Optimization Theory and Applications}, keywords = {Cooperative games; Timed token rotation protocols; Vector optimization}, note =
{UnivIS-Import:2015-03-09:Pub.2000.nat.dma.pama21.gameth}, pages = {463-474}, peerreviewed = {Yes}, title = {{Game}-{Theoretic} {Approach} to the {Optimization} of {FDDI} {Computer} {Networks}},
volume = {106}, year = {2000} } @article{faucris.123311144, abstract = {We analyze the subcritical gas flow through fan-shaped networks of pipes, that is, through tree-shaped networks with exactly
one node where more than two pipes meet. The gas flow in pipe networks is modeled by the isothermal Euler equations, a hyperbolic PDE system of balance laws. For this system we analyze stationary
states and classical nonstationary solutions locally around a stationary state on a finite time interval. Furthermore, we present a Lyapunov function and boundary feedback laws to stabilize a
fan-shaped network around a given stationary state. © 2011 Society for Industrial and Applied Mathematics.}, author = {Gugat, Martin and Hirsch-Dick, Markus and Leugering, Günter}, doi = {10.1137/
100799824}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, keywords = {Classical solutions; Critical length; Fan-shaped networks; Feedback law; Gas networks; Junction;
Lyapunov function; Networked hyperbolic systems; Riemann invariants}, note = {UnivIS-Import:2015-03-09:Pub.2011.nat.dma.zentr.gasflo}, pages = {2101-2117}, peerreviewed = {Yes}, title = {{Gas} {Flow}
in {Fan}-{Shaped} {Networks}: {Classical} {Solutions} and {Feedback} {Stabilization}}, url = {http://epubs.siam.org/sicon/resource/1/sjcodc/v49/i5/p2101{\_}s1}, volume = {49}, year = {2011} }
@article{faucris.110997084, abstract = {Necessary and sufficient optimality conditions are given for various optimality notions in set-valued optimization. These optimality conditions are given by
employing the generalized contingent epiderivative and the weak contingent epiderivative of the objective set-valued map and the set-valued map defining the constraints. The known Lagrange multiplier
rule and the so-called Zowe-Kurcyusz-Robinson (cf. Robinson, S.M. Stability Theory for Systems of Inequalities. II, Differentiable Nonlinear Systems. SIAM J. Numer. Anal. 1976, 13, 497-513. Zowe, J.;
Kurcyusz, S. Regularity and Stability for the Mathematical Programming Problem in Banach spaces, Appl. Math. Optim 1979, 5, 49-62.) regularity condition are extended using these differentiability
notions.}, author = {Jahn, Johannes and Khan, Akhtar Ali}, doi = {10.1081/NFA-120016271}, faupublication = {yes}, journal = {Numerical Functional Analysis and Optimization}, keywords = {Aubin
property; Contingent cone; Generalized contingent epiderivative; Lagrange multipliers; Optimality conditions; Regularity conditions; Set-valued optimization}, note =
{UnivIS-Import:2015-03-09:Pub.2002.nat.dma.pama21.genera}, pages = {807-831}, peerreviewed = {Yes}, title = {{Generalized} contingent epiderivatives in set-valued optimization: optimality
conditions}, volume = {23}, year = {2002} } @article{faucris.110298144, abstract = {Quasiconvexity for mappings is generalized in such a way that this notion gives the sufficiency of necessary
optimality conditions such as multiplier rules. One can show that it is the weakest type of generalized convexity notions in the sense that this generalized quasiconvexity holds if certain multiplier
rules are sufficient for optimality. It also yields the equivalence of local and global minima. The theory is applied to a multi-objective programming problem and a vector approximation problem.},
author = {Jahn, Johannes and Sachs, E.}, faupublication = {no}, journal = {SIAM Journal on Control and Optimization}, note = {UnivIS-Import:2015-03-05:Pub.1986.nat.dma.pama21.genera}, pages =
{306-322}, peerreviewed = {Yes}, title = {{Generalized} {Quasiconvex} {Mappings} and {Vector} {Optimization}}, volume = {24}, year = {1986} } @article{faucris.123196084, abstract = {Control over the
deformation behaviour that a cellular structure shows in response to imposed external forces is a requirement for the effective design of mechanical metamaterials, in particular those with negative
Poisson's ratio. This article sheds light on the old question of the relationship between geometric microstructure and mechanical response, by comparison of the deformation properties of
bar-and-joint frameworks with those of their realisation as a cellular solid made from linear-elastic material. For ordered planar tessellation models, we find a classification in terms of the number
of degrees of freedom of the framework model: first, in cases where the geometry uniquely prescribes a single deformation mode of the framework model, the mechanical deformation and Poisson's ratio
of the linearly-elastic cellular solid closely follow those of the unique deformation mode; the result is a bending-dominated deformation with negligible dependence of the effective Poisson's ratio
on the underlying material's Poisson's ratio and small values of the effective Young's modulus. Second, in the case of rigid structures or when geometric degeneracy prevents the bending-dominated
deformation mode, the effective Poisson's ratio is material-dependent and the Young's modulus (E) over tilde (cs) large. All analysed structures of this type have positive values of the Poisson's
ratio and large values of (E) over tilde (cs). Third, in the case, where the framework has multiple deformation modes, geometry alone does not suffice to determine the mechanical deformation. These
results clarify the relationship between mechanical properties of a linear-elastic cellular solid and its corresponding bar-and-joint framework abstraction. They also raise the question if, in
essence, auxetic behaviour is restricted to the geometry-guided class of bending-dominated structures corresponding to unique mechanisms, with inherently low values of the Young's modulus. (C) 2016
Elsevier Ltd. All rights reserved.}, author = {Mitschke, Holger and Schury, Fabian and Mecke, Klaus and Wein, Fabian and Stingl, Michael and Schroder-Turk, Gerd E.}, doi = {10.1016/
j.ijsolstr.2016.06.027}, faupublication = {yes}, journal = {International Journal of Solids and Structures}, keywords = {Cellular structures;Homogenisation;Floppy frameworks;Auxeticity;Mechanical
metamaterial}, pages = {1-10}, peerreviewed = {Yes}, title = {{Geometry}: {The} leading parameter for the {Poisson}'s ratio of bending-dominated cellular solids}, volume = {100}, year = {2016} }
@article{faucris.120144244, abstract = {We consider the problem of exactly controlling the states of the de St. Venant equations from a given constant state to another constant state by applying
nonlinear boundary controls. During this transition the solution stays in the class of C^1-solutions. There are no restrictions on the distance between the initial state and the target state, so our
result is a global controllability result for a nonlinear hyberbolic system. © 2003 Éditions scientifiques et médicales Elsevier SAS.}, author = {Leugering, Günter and Gugat, Martin}, doi = {10.1016/
S0294-1449(02)00004-5}, faupublication = {yes}, journal = {Annales de l'Institut Henri Poincaré - Analyse Non Linéaire}, keywords = {Characteristic form; de St. Venant equation; Global
controllability; Nonlinear hyperbolic system}, note = {UnivIS-Import:2015-03-09:Pub.2003.nat.dma.lama1.global}, pages = {1-11}, peerreviewed = {Yes}, title = {{Global} boundary controllability of the
de {St}. {Venant} equations between steady states}, volume = {20}, year = {2003} } @article{faucris.112052644, abstract = {We consider a sloped canal with friction that is governed by the
Saint-Venant system with source term. We show that starting sufficiently close to a stationary constant subcritical initial state, we can control the system in finite time to a state in a C^1
neighbourhood of any other stationary constant subcritical state by boundary control at the ends of the canal in such a way that during the process the system state remains continuously
differentiable. Moreover, we show that if the derivative of the initial state is sufficiently small, it can be steered to every stationary constant subcritical state in finite time. © 2008.}, author
= {Gugat, Martin and Leugering, Günter}, doi = {10.1016/j.anihpc.2008.01.002}, faupublication = {yes}, journal = {Annales de l'Institut Henri Poincaré - Analyse Non Linéaire}, keywords = {Friction;
Global controllability; Nonlinear hyperbolic system; Saint-Venant equation; Slope; Source term}, note = {UnivIS-Import:2015-03-09:Pub.2009.nat.dma.zentr.global}, pages = {257-270}, peerreviewed =
{Yes}, title = {{Global} boundary controllability of the {Saint}-{Venant} system for sloped canals with friction}, url = {http://www.sciencedirect.com/science?{\_}ob=ArticleURL&{\_}udi=
{\_}version=1&{\_}urlVersion=0&{\_}userid=616145&md5=31e90bfd85a87331d01e34c56914fa42}, volume = {26}, year = {2009} } @article{faucris.111290564, abstract = {We consider a tree-like network of open
channels with outflow at the root. Controls are exerted at the boundary nodes of the network except for the root. In each channel, the flow is modelled by the de St. Venant equations. The node
conditions require the conservation of mass and the conservation of energy. We show that the states of the system can be controlled within the entire network in finite time from a stationary
supercritical initial state to a given supercritical terminal state with the same orientation. During this transition, the states stay in the class of C^1-functions, so no shocks occur. Copyright ©
2004 John Wiley & Sons, Ltd.}, author = {Gugat, Martin and Leugering, Günter and Schmidt, E. J. P. Georg}, doi = {10.1002/mma.471}, faupublication = {yes}, journal = {Mathematical Methods in the
Applied Sciences}, keywords = {Global controllability; Network; Node conditions; St. Venant equations; Supercritical states}, note = {UnivIS-Import:2015-03-09:Pub.2004.nat.dma.zentr.global}, pages =
{781-802}, peerreviewed = {Yes}, title = {{Global} controllability between steady supercritical flows in channel networks}, url = {http://www3.interscience.wiley.com/cgi-bin/abstract/108561139/
ABSTRACT?CRETRY=1&SRETRY=0}, volume = {27}, year = {2004} } @article{faucris.112050004, abstract = {Optimality conditions are given for nonlinear bilevel vector optimization problems in infinite
dimensions. A new numerical method based on a multiobjective search algorithm with subdivision technique, is introduced for the determination of global solutions. Numerical results are presented for
low dimensional nonlinear problems. © 2009 Yokohama Publishers.}, author = {Jahn, Johannes and Gebhardt, E.}, faupublication = {yes}, journal = {Pacific Journal of Optimization}, keywords = {Bilevel
optimization; Global solvers; Vector optimization}, note = {UnivIS-Import:2015-03-09:Pub.2009.nat.dma.pama21.global}, pages = {387-401}, peerreviewed = {Yes}, title = {{Global} solver for nonlinear
bilevel vector optimization problems}, volume = {5}, year = {2009} } @article{faucris.112737284, abstract = {In this paper we observe the possibility to accelerate a search algorithm for
multiobjective optimization problems with help of a graphics processing unit. Besides an implementation we present test results for it and the conclusions that can be drawn from these results. © 2012
Springer Basel AG.}, author = {Limmer, Steffen and Fey, Dietmar and Jahn, Johannes}, doi = {10.1007/s11117-012-0156-x}, faupublication = {yes}, journal = {Positivity}, keywords = {CUDA; GPGPU;
Multiobjective search algorithm; Optimization; Parallelization}, note = {UnivIS-Import:2015-03-09:Pub.2012.nat.dma.pama21.gpuimp}, pages = {397-404}, peerreviewed = {Yes}, title = {{GPU}
implementation of a multiobjective search algorithm}, volume = {16}, year = {2012} } @misc{faucris.115282904, author = {Limmer, Steffen and Fey, Dietmar and Jahn, Johannes}, faupublication = {yes},
note = {UnivIS-Import:2016-06-30:Pub.2011.tech.IMMD.IMMD3.gpuimp}, peerreviewed = {automatic}, title = {{GPU} {Implementation} of a {Multiobjective} {Search} {Algorithm}}, url = {http://
www.opus.ub.uni-erlangen.de/opus/volltexte/2011/2538/}, year = {2011} } @inproceedings{faucris.121391204, address = {Aachen}, author = {Jahn, Johannes and Rathje, U.}, booktitle = {Multicriteria
Decision Making and Fuzzy Systems - Theory, Methods and Applications}, editor = {K.H. Küfer, H. Rommelfanger, C. Tammer, K. Winkler}, faupublication = {yes}, note =
{UnivIS-Import:2015-04-16:Pub.2006.nat.dma.pama21.graefy}, pages = {75 - 81}, peerreviewed = {unknown}, publisher = {Shaker Verlag}, title = {{Graef}-{Younes} {Method} with {Backward} {Iteration}},
year = {2006} } @incollection{faucris.118371924, address = {Wiesbaden}, author = {Jahn, Johannes}, booktitle = {Multi-Criteria- und Fuzzy-Systeme in Theorie und Praxis}, editor = {W. Habenicht, B.
Scheubrein, R. Scheubrein}, faupublication = {yes}, note = {UnivIS-Import:2015-04-20:Pub.2003.nat.dma.pama21.grundl}, pages = {37-71}, peerreviewed = {unknown}, publisher = {DUV Gabler Edition
Wissenschaft}, title = {{Grundlagen} der {Mengenoptimierung}}, year = {2003} } @article{faucris.114041664, author = {Rathmann, Wigand and Krug, Andreas}, faupublication = {no}, journal =
{Infoplaner}, pages = {36--37}, peerreviewed = {No}, title = {{Grundlagen} {FEM}: {Finite}-{Differenzen}-{Verfahren}}, year = {2004} } @article{faucris.120837024, author = {Rathmann, Wigand and Krug,
Andreas}, faupublication = {no}, journal = {Infoplaner}, pages = {44--45}, peerreviewed = {No}, title = {{Grundlagen} {FEM}: {Galerkin}-{Verfahren}}, year = {2005} } @article{faucris.119541004,
author = {Rathmann, Wigand and Gebald, Christoph}, faupublication = {no}, journal = {Infoplaner}, pages = {36--37}, peerreviewed = {No}, title = {{Grundlagen} {FEM}: {Ritz}-{Verfahren}}, volume = {1/
2003}, year = {2003} } @article{faucris.114041444, author = {Rathmann, Wigand and Krug, Andreas}, faupublication = {no}, journal = {Infoplaner}, pages = {38--39}, peerreviewed = {No}, title =
{{Grundlagen} {FEM}: {Trefftz}-{Verfahren}}, volume = {2/2003}, year = {2003} } @incollection{faucris.121857384, address = {Stuttgart}, author = {Peukert, Wolfgang and Schwarzer, Hans-Christoph and
Leugering, Günter}, booktitle = {Produktgestaltung in der Partikeltechnologie}, editor = {U. Teipel}, faupublication = {no}, isbn = {3-8167-6204-2}, keywords = {nano; particle; product;}, note =
{UnivIS-Import:2015-04-20:Pub.2002.tech.ITC.mechve.guidel}, pages = {-}, peerreviewed = {No}, publisher = {Fraunhofer-IRB Verlag}, series = {Produktgestaltung in der Partikeltechnologie.}, title =
{{Guidelines} for the production of nanoscaled particles and products}, year = {2002} } @inproceedings{faucris.244449075, abstract = {
Since several years an engineer and a mathematician offer the course "Simulation of transportation processes using Matlab" for master students of process and power engineering. This is an
interdisciplinary course where the students will get an elementary introduction to numerics and coding in Matlab/Octave, will repeat the modelling of heat flow and steady Rankine processes and
combine this in some Matlab scripts. This blended course already engaged the students to work independently but to meet every two weeks for discussion and get support in programming. For the two
weeks between the sessions we have prepared learning modules provided in our LMS ILIAS, which the students have to prepare. This is basis for the discussion in the presence meetings. In this learning
modules, e.g. the 1d heat transfer was introduced and in parallel the idea of the finite differences of first and second order. For this a loop back was done to the engineering mathematics courses 1
and 2.
All this we turned now into a pure online course, in particular the coding sessions. Some adjustments we have done, so we offered a weekly question hour. The main headline about all is: Getting the
students in to action.
At the end we have to state, that this year the students were more active and discussed the models much more deeply then before.
This course is held together with Micheal Wensing (Institute of Engineering Thermodynamics).
}, author = {Rathmann, Wigand}, booktitle = {Teaching and Learning of Mathematical and Statistical Computing Online}, date = {2020-09-21/2020-09-21}, faupublication = {yes}, peerreviewed = {No},
title = {{Guiding} students online in learning to code}, venue = {Online}, year = {2020} } @article{faucris.117906624, abstract = {The authors consider the problem of boundary feedback stabilization
of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H (2)-norm. To this end, an explicit Lyapunov function as a weighted and squared
H (2)-norm of a small perturbation of the stationary solution is constructed. The authors show that by a suitable choice of the boundary feedback conditions, the H (2)-exponential stability of the
stationary solution follows. Due to this fact, the system is stabilized over an infinite time interval. Furthermore, exponential estimates for the C (1)-norm are derived.}, author = {Gugat, Martin
and Leugering, Günter and Tamasoiu, Simona Oana and Wang, Ke}, doi = {10.1007/s11401-012-0727-y}, faupublication = {yes}, journal = {Chinese Annals of Mathematics Series B}, keywords = {Boundary
control;Feedback stabilization;Quasilinear hyperbolic system;Balance law;Gas dynamics;Isothermal Euler equations;Exponential stability;Lyapunov function;H-2-norm;Stationary state;Characteristic
variable; 76N25; 35L50; 93C20}, pages = {479-500}, peerreviewed = {Yes}, title = {{H} (2)-stabilization of the {Isothermal} {Euler} equations: a {Lyapunov} function approach}, url = {http://
link.springer.com/article/10.1007/s11401-012-0727-y}, volume = {33}, year = {2012} } @article{faucris.107378744, author = {Bischoff, Martin and Jahn, Johannes and Köbis, Elisabeth}, faupublication =
{yes}, journal = {Optimization}, pages = {361-380}, peerreviewed = {Yes}, title = {{Hard} uncertainties in multiobjective layout optimization of photovoltaic power plants}, volume = {66}, year =
{2017} } @article{faucris.122597244, abstract = {Dielectric-metal core-shell particles with morphologically tunable optical properties are highly promising candidates for applications ranging from
theranostics, energy harvesting and storage to pigments and sensors. Most structures of interest have, until now, been produced in small volume wet chemical batch approaches which are difficult to
scale. In extension to the growing interest in continuous flow process for the scalable synthesis of single phase nanomaterials, here we describe the formation of two-phase core-shell particles.
Specifically, the coating of silver onto colloidal silica particles using a static T-mixer approach is presented. The coating is achieved in a single reaction step in which careful control of educt
concentration and process conditions leads to nucleation and surface conformal growth of silver. Through two generations of reactor, good coating yield and process reproducibility are demonstrated.
In particular, the ability to control the characteristic time of mixing, as verified by the Villermaux-Dushmann reaction, was shown to be crucial in narrowing the range of morphologies obtained in
the product particles. Moreover we could tune, through process parameters, the degree of coating of the core particles, leading to patchy particles, Janus particles and even complete nanoshells.
Since silver nanostructures have a strong structure-dependent plasmon resonance, we could verify the coating morphologies via optical spectroscopy and electrodynamic simulations. Optical measurements
were also used in order to optimize, via trial and error, the continuous flow process parameters to produce a high coating yield. This opens up future potential for feedback-driven optimization of
particle coating. Finally, we demonstrate that the silver coatings can be morphologically stabilized, opening up the possibility of effective formulation for applications.}, author = {Meincke, Thomas
and Bao, Huixin and Pflug, Lukas and Stingl, Michael and Klupp Taylor, Robin}, doi = {10.1016/j.cej.2016.09.048}, faupublication = {yes}, journal = {Chemical Engineering Journal}, keywords =
{Continuous flow nanoparticle synthesis; Diffusion limited growth; Metal nanoshell; Microreactor; Patchy particle; Plasmon resonance}, pages = {89-100}, peerreviewed = {Yes}, title = {{Heterogeneous}
nucleation and surface conformal growth of silver nanocoatings on colloidal silica in a continuous flow static {T}-mixer}, volume = {308}, year = {2017} } @article{faucris.120887844, author = {Bley,
Karina and Semmler, Johannes and Rey, Marcel and Zhao, Chunjing and Martic, Nemanja and Klupp Taylor, Robin and Stingl, Michael and Vogel, Nicolas}, doi = {10.1002/adfm.201706965}, faupublication =
{yes}, journal = {Advanced Functional Materials}, keywords = {Colloidal lithography; Haze factor; Hierarchy; Nanophotonics; Transparent electrodes}, peerreviewed = {Yes}, title = {{Hierarchical}
{Design} of {Metal} {Micro}/{Nanohole} {Array} {Films} {Optimizes} {Transparency} and {Haze} {Factor}}, year = {2018} } @article{faucris.117906844, abstract = {We are concerned with the asymptotic
analysis of optimal control problems for 1-D partial differential equations defined on a periodic planar graph, as the period of the graph tends to zero. We focus on optimal control problems for
elliptic equations with distributed and boundary controls. Using approaches of the theory of homogenization we show that the original problem on the periodic graph tends to a standard linear
quadratic optimal control problem for a two-dimensional homogenized system, and its solution can be used as suboptimal controls for the original problem.}, author = {Kogut, Peter I. and Leugering,
Günter}, doi = {10.1051/cocv:2008037}, faupublication = {yes}, journal = {Esaim-Control Optimisation and Calculus of Variations}, keywords = {Optimal control; homogenization; elliptic equation;
periodic graph; 35B27; 35J25; 49J20; 93C20}, pages = {471-498}, peerreviewed = {Yes}, title = {{HOMOGENIZATION} {OF} {CONSTRAINED} {OPTIMAL} {CONTROL} {PROBLEMS} {FOR} {ONE}-{DIMENSIONAL} {ELLIPTIC}
{EQUATIONS} {ON} {PERIODIC} {GRAPHS}}, url = {http://www.esaim-cocv.org/articles/cocv/abs/2009/02/cocv0505/cocv0505.html}, volume = {15}, year = {2009} } @article{faucris.117907504, abstract = {We
consider an elliptic distributed quadratic optimal control problem with exact controllability constraints on a part of the domain which, in turn, is parametrized by a small parameter e. The quadratic
tracking type functional is defined on the remaining part of the domain. We thus consider a family of optimal control problems with state equality constraints. The purpose of this paper is to study
the asymptotic limit of the optimal control problems as the parameter e tends to zero. The analysis presented is in the spirit of the direct approach of the calculus of variations. This is achieved
in the framework of relaxed problems. We finally apply the procedure to an optimal control problem on a perforated domain with holes of critical size. It is shown that a strange term in the
terminology of Cioranescu and Murat (Prog. Nonlinear Diff. Eq. Appl., Vol. 31, Birkhauser-Verlag, Boston, 1997, pp. 49-93) appears in the limiting homogenized problem.}, author = {Kogut, Peter I. and
Leugering, Günter}, doi = {10.3233/ASY-2008-0871}, faupublication = {yes}, journal = {Asymptotic Analysis}, keywords = {controllability constraints;relaxed problems;gamma(Delta) and
Gamma-convergence;perforated domains}, month = {Jan}, pages = {229-249}, peerreviewed = {Yes}, title = {{Homogenization} of {Dirichlet} optimal control problems with exact partial controllability
constraints}, url = {http://content.iospress.com/articles/asymptotic-analysis/asy871}, volume = {57}, year = {2008} } @article{faucris.117908604, abstract = {We consider the optimal control problems
for the linear elliptic equations in perforated domains. Each components of mathematical description of such optimal control problem depends on small parameter epsilon. Problems of this type appear
in sensitivity analysis, perturbation theory, and homogenization of heterogenous material. We study the problem of passing to the limit within the framework of variational S-convergence. We derive
conditions under which the so-called fictitious limiting optimal control problem can be made explicit.}, author = {Kogut, Peter I. and Leugering, Günter}, faupublication = {no}, journal = {Asymptotic
Analysis}, keywords = {homogenization; S-convergence; optimal control; variable domains}, pages = {37-72}, peerreviewed = {Yes}, title = {{Homogenization} of optimal control problems in variable
domains. {Principle} of the fictitious homogenization}, url = {http://content.iospress.com/articles/asymptotic-analysis/asy438}, volume = {26}, year = {2001} } @inproceedings{faucris.111241944,
author = {Rathmann, Wigand and Vogel, Frank and Landes, Hermann}, booktitle = {21. CAD-FEM Users' Meeting}, faupublication = {no}, peerreviewed = {unknown}, publisher = {CAD-FEM GmbH Grafing}, title
= {{How} to use {Diffpack} complementary to other {FEM} packages}, year = {2003} } @phdthesis{faucris.123772044, author = {Hante, Falk}, faupublication = {yes}, peerreviewed = {automatic}, school =
{Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{Hybrid} {Dynamics} {Comprising} {Modes} {Governed} by {Partial} {Differential} {Equations}: {Modeling}, {Analysis} and {Control} for
{Semilinear} {Hyperbolic} {Systems} in {One} {Space} {Dimension}}, url = {http://nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bvb:29-opus-19356}, year = {2010} } @article{faucris.106971304, author
= {Leugering, Günter}, doi = {10.1002/gamm.201210008}, faupublication = {yes}, journal = {GAMM-Mitteilungen}, pages = {108-109}, peerreviewed = {unknown}, title = {{Identification}, optimization and
control for modern technologies}, volume = {35}, year = {2012} } @inproceedings{faucris.235431372, author = {Rathmann, Wigand and Copado Mejías, Jesús}, booktitle = {International STACK user group
meeting (2. STACK Conference)}, date = {2019-04-30/2019-04-30}, faupublication = {yes}, peerreviewed = {unknown}, title = {{Implementing} {STACK} questions concerning integration techniques combined
with the usage of {STACK} in {ILIAS}}, url = {https://www.maths.ed.ac.uk/~csangwin/2019-STACK/2019-Rathmann.pdf}, venue = {International Center for Mathematical Sciences, Edingurgh}, year = {2019} }
@inproceedings{faucris.120766844, author = {Rathmann, Wigand and Vogel, Frank}, booktitle = {20. CAD-FEM Users' Meeting}, faupublication = {no}, peerreviewed = {No}, publisher = {CAD-FEM GmbH
Grafing}, title = {{Integration} of {Diffpack} with {FEM} {Application}}, volume = {2}, year = {2002} } @article{faucris.121355564, author = {Hopfgartner, Christian and Scholz, Ingo and Gugat, Martin
and Leugering, Günter and Hornegger, Joachim}, faupublication = {yes}, journal = {ICGST International Journal on Graphics, Vision and Image Processing}, pages = {27-37}, peerreviewed = {unknown},
title = {{Intensity}-based 3-{D} {Reconstruction} with {Non}-linear {Optimization}}, url = {http://www5.informatik.uni-erlangen.de/Forschung/Publikationen/2010/Hopfgartner10-IRW.pdf}, volume =
{10.0}, year = {2010} } @misc{faucris.109715804, abstract = {New images of a three-dimensional scene can be generated from known image sequences using lightfields. To get high quality images, it is
important to have accurate information about the structure of the scene. In order to optimize this information, we define a residual-function. This function represents the difference between an
image, rendered in a known view from neighboured images and the original image at the same position. In order to get optimal results, we minimize the residual-function by defining a nonlinear
least-squares problem, which is solved by an appropriate optimization method. We use a nonmonotone variant of the Levenberg-Marquardt method.}, author = {Hopfgartner, Christian and Scholz, Ingo and
Gugat, Martin and Leugering, Günter and Hornegger, Joachim}, faupublication = {yes}, keywords = {Nonlinear Optimization; Imaging; Rendering; Nonmonotone Levenberg-Marquardt; 90C30; 68U10; 94A08},
peerreviewed = {automatic}, title = {{Intensity} based {Three}-{Dimensional} {Reconstruction} with {Nonlinear} {Optimization}}, url = {http://www.am.uni-erlangen.de/de/preprints2000.html}, year =
{2007} } @article{faucris.122365364, abstract = {Mesocrystalline particles have been recognized as a class of multifunctional materials with potential applications in different fields. However, the
internal organization of nanocomposite mesocrystals and its influence on the final properties have not yet been investigated. In this paper, a novel strategy based on electrodynamic simulations is
developed to shed light on how the internal structure of mesocrystals influences their optical properties. In a first instance, a unified design protocol is reported for the fabrication of hematite/
PVP particles with different morphologies such as pseudo-cubes, rods-like and apple-like structures and controlled particle size distributions. The optical properties of hematite/PVP mesocrystals are
effectively simulated by taking their aggregate and nanocomposite structure into consideration. The superposition T-Matrix approach accounts for the aggregate nature of mesocrystalline particles and
validate the effective medium approximation used in the framework of the Mie theory and electromagnetic simulation such as Finite Element Method. The approach described in our paper provides the
framework to understand and predict the optical properties of mesocrystals and more general, of hierarchical nanostructured particles.}, author = {Distaso, Monica and Zhuromskyy, Oleksandr and
Seemann, Benjamin and Pflug, Lukas and Mackovic, Mirza and Encina, Ezequiel Roberto and Klupp Taylor, Robin and Müller, Rolf and Leugering, Günter and Spiecker, Erdmann and Peschel, Ulf and Peukert,
Wolfgang}, doi = {10.1016/j.jqsrt.2016.12.028}, faupublication = {yes}, journal = {Journal of Quantitative Spectroscopy & Radiative Transfer}, keywords = {Effective medium approximation; FEM;
Maxwell-Garnett theory; Mesocrystals; Scattering; T-Matrix}, pages = {369-382}, peerreviewed = {Yes}, title = {{Interaction} of light with hematite hierarchical structures: {Experiments} and
simulations}, volume = {189}, year = {2017} } @article{faucris.118515584, abstract = {In this paper, we consider a model for precipitation experiments based on the population balance equation. The
study revealed a high sensitivity of the system with respect to the modeling of intrinsic parameters, motivating a comprehensive validation of the estimates. In the forward simulation the impact of
the influencing parameters including surface energy, nucleus size and distribution is investigated. Subsequently we construct a simplified model of the precipitation process in such a way that it is
orbitally flat in terms of control theory, which enables the inverse calculation of the parameters. The numerical results of the inverse simulation for the interfacial energy have been compared to a
physical model. The possibility of solving the inverse problem provides a promising way of estimating hardly measurable quantities for more complex molecules.}, author = {Vassilev, Vassil and
Gröschel, Michael and Schmid, Hans-Joachim and Peukert, Wolfgang and Leugering, Günter}, doi = {10.1016/j.ces.2009.12.014}, faupublication = {yes}, journal = {Chemical Engineering Science}, pages =
{2183--2189}, peerreviewed = {Yes}, title = {{Interfacial} energy estimation in a precipitation reaction using the flatness based control of the moment trajectories}, volume = {65}, year = {2010} }
@book{faucris.118438584, abstract = {International Series of Numerical Mathematics is open to all aspects of numerical mathematics. Some of the topics of particular interest include free boundary
value problems for differential equations, phase transitions, problems of optimal control and optimization, other nonlinear phenomena in analysis, nonlinear partial differential equations, efficient
solution methods, bifurcation problems and approximation theory. When possible, the topic of each volume is discussed from three different angles, namely those of mathematical modeling, mathematical
analysis, and numerical case studies.}, address = {Basel}, editor = {Leugering, Günter and Hintermüller, Michael}, faupublication = {yes}, note =
{UnivIS-Import:2015-05-08:Pub.2007.nat.dma.zentr.optima}, publisher = {Birkhäuser}, title = {{International} {Series} of {Numerical} {Mathematics}}, year = {2007} } @book{faucris.108768264, address =
{Berlin}, author = {Jahn, Johannes}, edition = {3}, faupublication = {yes}, note = {UnivIS-Import:2015-04-02:Pub.1994.nat.dma.pama21.introd}, pages = {249}, publisher = {Springer}, title =
{{Introduction} to the {Theory} of {Nonlinear} {Optimization}}, year = {2007} } @article{faucris.228941119, abstract = {Optimization tasks under uncertain conditions abound in many real-life
applications. Whereas solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches
are usually used in case solutions are sought that are feasible for all realizations of uncertainties within some pre-defined uncertainty set. As many applications contain different types of
uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints as its appears in optimization problems under uncertainty.
Focussing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a
stationary gas network. We furthermore present approaches for their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists
in determining controls such that a certain robustness measure remains below some given upper bound, with high probabilit}, author = {Adelhütte, Dennis and Aßmann, Denis and Gonzàlez Grandòn, Tatiana
and Gugat, Martin and Heitsch, Holger and Liers, Frauke and Henrion, René and Nitsche, Sabrina and Schultz, Rüdiger and Stingl, Michael and Wintergerst, David}, doi = {10.1007/s10013-020-00434-y},
faupublication = {yes}, journal = {Vietnam Journal of Mathematics}, keywords = {Stabilization; Wave equation; Feedback; Robust optimization; Probabilistic constraints; Probust; Karhunen–Loève},
peerreviewed = {Yes}, title = {{Joint} model of probabilistic/robust (probust) constraints with application to gas network optimization}, url = {https://link.springer.com/article/10.1007/
s10013-020-00434-y}, year = {2020} } @article{faucris.108502284, author = {Jahn, Johannes}, faupublication = {yes}, journal = {Journal of Optimization Theory and Applications}, pages = {707-725},
peerreviewed = {Yes}, title = {{Karush}-{Kuhn}-{Tucker} {Conditions} in {Set} {Optimization}}, volume = {172}, year = {2017} } @inproceedings{faucris.117327364, abstract = {In this paper, the problem
to control a finite string to the zero state in finite time from a given initial state by controlling the state at the two boundary points is considered. The corresponding optimal control problem
where the objective function is the L^l-norm of the controls is solved in the sense that the controls that are successful and minimize at the same time the objective function are determined as
functions of the initial state. © 2006 Springer-Verlag Berlin Heidelberg.}, address = {Berlin}, author = {Gugat, Martin}, booktitle = {Recent Advances in Optimization}, date = {2004-09-20/
2004-09-24}, doi = {10.1007/3-540-28258-0{\_}10}, editor = {Alberto Seeger}, faupublication = {yes}, isbn = {978-3-540-28257-0}, keywords = {L1-optimal boundary control, exact control, wave equation,
Dirichlet boundary conditions}, note = {UnivIS-Import:2015-04-16:Pub.2006.nat.dma.zentr.l1opti}, pages = {149-162}, publisher = {Springer Verlag}, series = {Lecture Notes in Economics and
Mathematical Systems}, title = {{L1}-{Optimal} boundary control of a string to rest in finite time}, venue = {Avignon}, volume = {563}, year = {2006} } @article{faucris.215204958, abstract = {This
paper presents a new Lagrange theory of discrete-continuous conic optimization in an infinite dimensional setting. The following questions are answered for discrete-continuous optimization problems:
how to define a Lagrange functional, how Karush-Kuhn-Tucker conditions look like, and which duality results can be obtained? This approach is based on new separation theorems for discrete sets, which
are also given in this paper. The developed theory is finally applied to problems of discrete-continuous semidefinite and copositive optimization.}, author = {Jahn, Johannes and Knossalla, Martin},
doi = {10.23952/jnva.2.2018.3.07}, faupublication = {yes}, journal = {Journal of Nonlinear and Variational Analysis}, month = {Jan}, note = {CRIS-Team WoS Importer:2019-04-02}, pages = {317-342},
peerreviewed = {Yes}, title = {{LAGRANGE} {THEORY} {OF} {DISCRETE}-{CONTINUOUS} {NONLINEAR} {OPTIMIZATION}}, volume = {2}, year = {2018} } @article{faucris.112053304, abstract = {A Lavrentiev
prox-regularization method for optimal control problems with point-wise state constraints is introduced where both the objective function and the constraints are regularized. The convergence of the
controls generated by the iterative Lavrentiev prox-regularization algorithm is studied. For a sequence of regularization parameters that converges to zero, strong convergence of the generated
control sequence to the optimal control is proved. Due to the proxcharacter of the proposed regularization, the feasibility of the iterates for a given parameter can be improved compared with the
non-prox Lavrentiev-Regularization. Mathematical subject classification: 49J20, 49M37. © 2009 Sociedade Brasileira de Matemática Aplicada e Computacional.}, author = {Gugat, Martin}, doi = {10.1590/
S1807-03022009000200006}, faupublication = {yes}, journal = {Computational and Applied Mathematics}, keywords = {Convergence; Feasibility; Lavrentiev regularization; Optimal control; PDE constrained
optimization; Pointwise state constraints; Prox regularization; 49J20; 49M37}, note = {UnivIS-Import:2015-03-09:Pub.2009.nat.dma.zentr.lavren}, pages = {231-257}, peerreviewed = {unknown}, title =
{{Lavrentiev}-prox-regularization for optimal control of {PDEs} with state constraints}, url = {http://www.scielo.br/scielo.php?script=sci{\_}abstract&pid=S1807-03022009000200006&lng=en&nrm=iso&tlng=
en}, volume = {28}, year = {2009} } @article{faucris.119935904, abstract = {For optimal control problems with ordinary differential equations where the L-infinity-norm of the control is minimized,
often bang-bang principles hold. For systems that are governed by a hyperbolic partial differential equation, the situation is different: even if a weak form of the bang-bang principle still holds
for the wave equation, it implies no restriction on the form of the optimal control. To illustrate that for the Dirichlet boundary control of the wave equation in general not even feasible controls
of bang-bang type exist, we examine the states that can be reached by bang-bang-off controls, that is controls that are allowed to attain only three values: Their maximum and minimum values and the
value zero. We show that for certain control times, the difference between the initial and the terminal state can only attain a finite number of values. For the problems of optimal exact and
approximate boundary control of the wave equation where the L-infinity-norm of the control is minimized, we introduce dual problems and present the weak form of a bang-bang principle, that states
that the values of L-infinity-norm minimal controls are constrained by the sign of the dual solutions. Since these dual solutions are in general given as measures, this is no restriction on the form
of the control function: the dual solution may have a finite support, and when the dual solution vanishes, the control is allowed to attain all values from the interval between the two extremal
control values.}, author = {Gugat, Martin and Leugering, Günter}, doi = {10.1051/cocv:2007044}, faupublication = {yes}, journal = {Esaim-Control Optimisation and Calculus of Variations}, keywords =
{optimal control of pdes;optimal boundary control;wave equation;bang-bang;bang-bang-off;dual problem;dual solutions;L-infinity;measures; 49K20; 35L05}, pages = {254-283}, peerreviewed = {Yes}, title
= {{L}-infinity-norm minimal control of the wave equation: {On} the weakness of the bang-bang principle}, url = {http://www.esaim-cocv.org/articles/cocv/abs/2008/02/cocv0585/cocv0585.html}, volume =
{14}, year = {2008} } @article{faucris.120961984, author = {Gugat, Martin and Hante, Falk}, doi = {10.1007/s00498-016-0183-4}, faupublication = {yes}, journal = {Mathematics of Control Signals and
Systems}, keywords = {Parametric optimal control; Parametric switching control; Parametric optimization; Sensitivity; Mixed-integer optimal control problems; Optimal value function; Lipschitz
continuity;}, peerreviewed = {Yes}, title = {{Lipschitz} {Continuity} of the {Value} {Function} in {Mixed}-{Integer} {Optimal} {Control} {Problems}}, volume = {29}, year = {2017} } @incollection
{faucris.222281916, author = {Hänsel, Matthias and Meßner, Arthur and Engel, Ulf and Geiger, Manfred}, booktitle = {Localized Damage II, Vol.2: Computational Methods in Fracture Mechanics},
faupublication = {yes}, note = {LFT Import::2019-07-15 (481)}, pages = {263-282}, peerreviewed = {unknown}, title = {{Local} {Energy} {Approach} - an {Advanced} {Model} for {FEM}-{Simulation} of
{Multiaxial} {Fatigue}}, year = {1992} } @article{faucris.290132175, abstract = {We study the long-time behavior of the unique weak solution of a nonlocal regularization of the (inviscid) Burgers'
equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is
given by the "N -wave'' entropy solution of the Burgers' equation. The key ingredients of the proof are a suitable scaling argument and a nonlocal Oleinik-type estimate.
}, author = {Coclite, Giuseppe Maria and De Nitti, Nicola and Keimer, Alexander and Pflug, Lukas and Zuazua Iriondo, Enrique}, doi = {10.1088/1361-6544/acf01d}, faupublication = {yes}, journal =
{Nonlinearity}, keywords = {Nonlocal conservation laws; nonlocal flux; Burgers’ equation; approximation of local conservation laws; N-waves; source-type solutions; entropy solutions}, peerreviewed =
{Yes}, title = {{Long}-time convergence of a nonlocal {Burgers}' equation towards the local {N}-wave}, url = {https://iopscience.iop.org/article/10.1088/1361-6544/acf01d}, year = {2023} }
@incollection{faucris.107023664, address = {München}, author = {Jahn, Johannes and Dupre, R. and Huckert, K.}, booktitle = {Ausgewählte Operations Research Software in FORTRAN}, editor = {H. Späth},
faupublication = {no}, note = {UnivIS-Import:2015-04-17:Pub.1979.nat.dma.pama21.lsungl}, pages = {9-29}, peerreviewed = {unknown}, publisher = {Oldenbourg}, title = {{Lösung} linearer
{Vektormaximumprobleme} durch das {STEM}-{Verfahren}}, year = {1979} } @incollection{faucris.118332544, address = {München}, author = {Jahn, Johannes}, booktitle = {Ausgewählte Operations Research
Software in FORTRAN}, editor = {H. Späth}, faupublication = {no}, note = {UnivIS-Import:2015-04-17:Pub.1979.nat.dma.pama21.lsungn}, pages = {30-50}, peerreviewed = {unknown}, publisher =
{Oldenbourg}, title = {{Lösung} nichtlinearer {Optimierungsprobleme} mit {Nebenbedingungen}}, year = {1979} } @article{faucris.111517164, abstract = {We study problems of boundary controllability
with minimal L -norm (p ∈ [2, ∞]) for the one-dimensional wave equation, where the state is controlled at both boundaries through Dirichlet or Neumann conditions. The problem is to reach a given
terminal state and velocity in a given finite time, while minimizing the L-norm of the controls. We give necessary and sufficient conditions for the solvability of this problem. We show as follows
how this infinite-dimensional optimization problem can be transformed into a problem which is much simpler: The feasible set of the transformed problem is described by a finite number of simple
pointwise equality constraints for the control function in the Dirichlet case while, in the Neumann case, an additional integral equality constraint appears. We provide explicit complete solutions of
the problems for all p ∈[2, ∞] in the Dirichlet case and solutions for some typical examples in the Neumann case. © 2005 Society for Industrial and Applied Mathematics.}, author = {Gugat, Martin and
Leugering, Günter and Sklyar, Gregori}, doi = {10.1137/S0363012903419212}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, keywords = {Analytic solution; Boundary
control; Controllability; Distributed parameter systems; Optimal control; Robust optimization; Sensitivity; State constraints; Test examples; Wave equation}, note =
{UnivIS-Import:2015-03-09:Pub.2005.nat.dma.lama1.lpopti}, pages = {49-74}, peerreviewed = {Yes}, title = {{Lp}-optimal boundary control for the wave equation,}, volume = {44}, year = {2005} }
@article{faucris.124194884, abstract = {Commonly applied models to study vocal fold vibrations in combination with air flow distributions are self-sustained physical models of the larynx consisting
of artificial silicone vocal folds. Choosing appropriate mechanical parameters and layer geometries for these vocal fold models while considering simplifications due to manufacturing restrictions is
difficult but crucial for achieving realistic behavior. In earlier work by Schmidt et al. [J. Acoust. Soc. Am. 129, 2168-2180 (2011)], the authors presented an approach in which material parameters
of a static numerical vocal fold model were optimized to achieve an agreement of the displacement field with data retrieved from hemilarynx experiments. This method is now generalized to a fully
transient setting. Moreover in addition to the material parameters, the extended approach is capable of finding optimized layer geometries. Depending on chosen material restriction, significant
modifications of the reference geometry are predicted. The additional flexibility in the design space leads to a significantly more realistic deformation behavior. At the same time, the predicted
biomechanical and geometrical results are still feasible for manufacturing physical vocal fold models consisting of several silicone layers. As a consequence, the proposed combined experimental and
numerical method is suited to guide the construction of physical vocal fold models. (C) 2013 Acoustical Society of America.}, author = {Schmidt, Bastian and Leugering, Günter and Stingl, Michael and
Hüttner, Björn and Agaimy, Abbas and Döllinger, Michael}, doi = {10.1121/1.4812253}, faupublication = {yes}, journal = {Journal of the Acoustical Society of America}, pages = {1261-1270},
peerreviewed = {Yes}, title = {{Material} and shape optimization for multi-layered vocal fold models using transient loadings}, volume = {134}, year = {2013} } @article{faucris.202370051, abstract =
{A class of algorithms for the solution of discrete material optimization problems in electromagnetic applications is discussed. The idea behind the algorithm is similar to that of the sequential
programming. However, in each major iteration a model is established on the basis of an appropriately parametrized material tensor. The resulting nonlinear parametrization is treated on the level of
the sub-problem, for which, globally optimal solutions can be computed due to the block separability of the model. Although global optimization of non-convex design problems is generally prohibitive,
a well chosen combination of analytic solutions along with standard global optimization techniques leads to a very efficient algorithm for most relevant material parametrizations. A global
convergence result for the overall algorithm is established. The effectiveness of the approach in terms of both computation time and solution quality is demonstrated by numerical examples, including
the optimal design of cloaking layers for a nano-particle and the identification of multiple materials with different optical properties in a matrix.}, author = {Semmler, Johannes and Pflug, Lukas
and Stingl, Michael}, doi = {10.1137/17M1127569}, faupublication = {yes}, journal = {SIAM Journal on Scientific Computing}, keywords = {Discrete optimization; Global optimization; Electromagnetic
scattering; Inverse problems; Helmholtz equation; Optical properties; Material optimization; Sequential programming}, pages = {B85-B109}, peerreviewed = {Yes}, title = {{Material} optimization in
transverse electromagnetic scattering applications}, volume = {40}, year = {2018} } @article{faucris.117640644, abstract = {Today, the prevention and treatment of voice disorders is an
ever-increasing health concern. Since many occupations rely on verbal communication, vocal health is necessary just to maintain one's livelihood. Commonly applied models to study vocal fold
vibrations and air flow distributions are self sustained physical models of the larynx composed of artificial silicone vocal folds. Choosing appropriate mechanical parameters for these vocal fold
models while considering simplifications due to manufacturing restrictions is difficult but crucial for achieving realistic behavior. In the present work, a combination of experimental and numerical
approaches to compute material parameters for synthetic vocal fold models is presented. The material parameters are derived from deformation behaviors of excised human larynges. The resulting
deformations are used as reference displacements for a tracking functional to be optimized. Material optimization was applied to three-dimensional vocal fold models based on isotropic and
transverse-isotropic material laws, considering both a layered model with homogeneous material properties on each layer and an inhomogeneous model. The best results exhibited a transversal-isotropic
inhomogeneous (i.e., not producible) model. For the homogeneous model (three layers), the transversal-isotropic material parameters were also computed for each layer yielding deformations similar to
the measured human vocal fold deformations. (C) 2011 Acoustical Society of America. [DOI:10.1121/1.3543988]}, author = {Schmidt, Bastian and Stingl, Michael and Leugering, Günter and Berry, David A.
and Döllinger, Michael}, doi = {10.1121/1.3543988}, faupublication = {yes}, journal = {Journal of the Acoustical Society of America}, pages = {2168-2180}, peerreviewed = {Yes}, title = {{Material}
parameter computation for multi-layered vocal fold models}, volume = {129}, year = {2011} } @incollection{faucris.119529784, address = {Berlin}, author = {Jahn, Johannes}, booktitle = {Essays and
Surveys on Multiple Criteria Decision Making}, editor = {P. Hansen}, faupublication = {no}, note = {UnivIS-Import:2015-04-17:Pub.1983.nat.dma.pama21.mathem}, pages = {177-186}, peerreviewed =
{unknown}, publisher = {Springer}, series = {Lecture Notes in Economics and Mathematical Systems}, title = {{Mathematical} {Applications} of {MCDM}: {Vector} {Approximation} and {Cooperative}
{Differential} {Games}}, volume = {209}, year = {1983} } @book{faucris.115716084, abstract = {Water supply- and drainage systems and mixed water channel systems are networks whose high dynamic is
determined and/or affected by consumer habits on drinking water on the one hand and by climate conditions, in particular rainfall, on the other hand. According to their size, water networks consist
of hundreds or thousands of system elements. Moreover, different types of decisions (continuous and discrete) have to be taken in the water management. The networks have to be optimized in terms of
topology and operation by targeting a variety of criteria. Criteria may for example be economic, social or ecological ones and may compete with each other. The development of complex model systems
and their use for deriving optimal decisions in water management is taking place at a rapid pace. Simulation and optimization methods originating in Operations Research have been used for several
decades; usually with very limited direct cooperation with applied mathematics. The research presented here aims at bridging this gap, thereby opening up space for synergies and innovation. It is
directly applicable for relevant practical problems and has been carried out in cooperation with utility and dumping companies, infrastructure providers and planning offices. A close and direct
connection to the practice of water management has been established by involving application-oriented know-how from the field of civil engineering. On the mathematical side all necessary disciplines
were involved, including mixed-integer optimization, multi-objective and facility location optimization, numerics for cross-linked dynamic transportation systems and optimization as well as control
of hybrid systems. Most of the presented research has been supported by the joint project "Discret continuous optimization of dynamic water systems" of the federal ministry of education and research
(BMBF).}, author = {Martin, Alexander and Klamroth, Kathrin and Lang, Jens and Leugering, Günter and Morsi, Antonio and Oberlack, Martin and Ostrowski, Manfred and Rosen, Roland}, faupublication =
{yes}, peerreviewed = {Yes}, publisher = {Birkhäuser}, series = {International Series of Numerical Mathematics}, title = {{Mathematical} {Optimization} of {Water} {Networks}}, volume = {162}, year =
{2012} } @book{faucris.108699404, address = {Frankfurt}, author = {Jahn, Johannes}, faupublication = {no}, note = {UnivIS-Import:2015-04-02:Pub.1986.nat.dma.pama21.mathem}, pages = {310}, publisher =
{Peter Lang}, title = {{Mathematical} {Vector} {Optimization} in {Partially} {Ordered} {Linear} {Spaces}}, year = {1986} } @inproceedings{faucris.244448530, abstract = {Das Sommersemester 2020 hat
eine neue Notwendigkeit digitaler Infrastruktur inder universitären Lehre gezeigt. In diesem Beitrag wird über den Übergang voneinem Präsenzunterricht mit Übungen und Korrektur von Hausaufgaben zu
einerOnlinelehre mit Präsenz berichtet. Zentraler "Backbone" waren die Elemente, dieILIAS in Kursen zur Verfügung stellt, wie Lernmodule, Sitzungen, Gestaltung derSeiten. Zentraler Punkt für den
Austausch waren das Übungsobjekt undLive-Votings als eines der wichtigen Elemente für die Interaktion und Aktivierung inden Online-Sitzungen. Die Auswertung der Live-Votings wurde
anschließenddokumentiert und der Stoff gemeinsam weiterentwickelt. Somit wurde versucht,die Vorteile der Präsenzlehre in den virtuellen Raum mitzunehmen. Um einblended Learning zu unterstützen, wurde
das Selbststudium gefordert undgefördert. Zudem werden Umfragen unter den Studierenden vorgestellt, die überdie Schwierigkeiten und Wünsche in Präsenz- und Onlinelehre Auskunft gebe}, author =
{Rathmann, Wigand}, booktitle = {19. International ILIAS Conference}, date = {2020-09-10}, faupublication = {yes}, peerreviewed = {No}, title = {{Mathematik} – {Präsenzlehre} online}, url = {https://
docu.ilias.de/goto{\_}docu{\_}file{\_}9239{\_}download.html}, venue = {Online}, year = {2020} } @inproceedings{faucris.106458924, author = {Rathmann, Wigand}, booktitle = {1. Tag der Lehre}, date =
{2015-12-11}, faupublication = {yes}, peerreviewed = {No}, title = {{Mathematische} {Fragestellungen} online realisieren ({Vortrag} beim 1. {Tag} der {Lehre} an der {FAU})}, url = {https://youtu.be/
GEvyM2vhm{\_}E?iframe=true}, venue = {FAU}, year = {2015} } @article{faucris.117696964, author = {Domschke, Pia and Groß, Martin and Hante, Falk and Hiller, Benjamin and Schewe, Lars and Schmidt,
Martin}, faupublication = {yes}, journal = {Gas und Wasserfach, Gas, Erdgas}, pages = {880-885}, peerreviewed = {No}, title = {{Mathematische} {Modellierung}, {Simulation} und {Optimierung} von
{Gastransportnetzwerken}}, url = {https://www.di-verlag.de/de/Zeitschriften/gwf-Gas-Erdgas/2015/11/Mathematische-Modellierung-Simulation-und-Optimierung-von-Gastransportnetzwerken}, volume = {156},
year = {2015} } @article{faucris.107409984, abstract = {We consider optimal control problems for linear degenerate elliptic equations with mixed boundary conditions. In particular, we take the
matrix-valued coeffcients A(x) of such systems as controls in L1(;RN(N+1) 2 ). One of the important features of the admissible controls is the fact that eigenvalues of the coeffcient matrices may
vanish in Equations of this type may exhibit non-uniqueness of weak solutions. Using the concept of convergence in variable spaces and following the direct method in the Calculus of variations, we
establish the solvability of this optimal control problem in the class of weak admissible solutions. © European Mathematical Society.}, author = {Kogut, Peter I. and Leugering, Günter}, doi =
{10.4171/ZAA/1493}, faupublication = {yes}, journal = {Zeitschrift für Analysis und ihre Anwendungen}, keywords = {Degenerate elliptic equations;control in coefficients;weighted Sobolev spaces;
Lavrentieff phenomenon;direct method in the Calculus of Variations}, month = {Jan}, pages = {433-456}, peerreviewed = {Yes}, title = {{Matrix}-{Valued} {L1}-{Optimal} {Controls} in the {Coefficients}
of linear elliptic problems}, volume = {32}, year = {2013} } @article{faucris.117641744, abstract = {We present a thorough investigation of the mechanical behaviour of a non-stochastic cellular
auxetic structure. A combination of experimental and numerical methods is used to gain a deeper understanding of the mechanical behaviour and its dependence on the geometric properties of the
cellular structure. The experimental samples are built from Ti-6Al-4V using selective electron beam melting, an additive manufacturing process giving the possibility to vary the geometry of the
structure in a highly controlled manner. The use of finite element simulations and mathematical homogenisation allows us also to investigate off-axis properties of the cellular material. This leads
to a more comprehensive understanding of the mechanical behaviour of the auxetics. Ultimately, the gained knowledge can be used to tailor auxetic materials to specific applications.}, author =
{Schwerdtfeger, Jan and Schury, Fabian and Stingl, Michael and Wein, Fabian and Singer, Robert and Körner, Carolin}, doi = {10.1002/pssb.201084211}, faupublication = {yes}, journal = {physica status
solidi (b)}, keywords = {auxetics;cellular materials;electron beam melting;finite element method;metals}, pages = {1347-1352}, peerreviewed = {Yes}, title = {{Mechanical} characterisation of a
periodic auxetic structure produced by {SEBM}}, volume = {249}, year = {2012} } @article{faucris.106725564, abstract = {
Typically, exact information of the whole subdifferential is not available for intrinsically nonsmooth objective functions such as for marginal functions. Therefore, the semismoothness of the
objective function cannot be proved or is even violated. In particular, in these cases standard nonsmooth methods cannot be used. In this paper, we propose a new approach to develop a converging
descent method for this class of nonsmooth functions. This approach is based on continuous outer subdifferentials introduced by us. Further, we introduce on this basis a conceptual optimization
algorithm and prove its global convergence. This leads to a constructive approach enabling us to create a converging descentmethod. Within the algorithmic framework, neither semismoothness nor
calculation of exact subgradients are required. This is in contrast to other approaches which are usually based on the assumption of semismoothness of the
objective function.
}, author = {Knossalla, Martin}, doi = {10.1080/02331934.2018.1426579}, faupublication = {yes}, journal = {Optimization}, keywords = {Mathematical programming; nonsmooth optimization; parametric
optimization; variational analysis and optimization}, month = {Jan}, pages = {1-21}, peerreviewed = {Yes}, title = {{Minimization} of marginal functions in mathematical programming based on
continuous outer subdifferentials}, year = {2018} } @article{faucris.120013784, abstract = {We study the transient optimization of gas transport networks including both discrete controls due to
switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This
combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable
Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of
these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using
general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.}, author = {Gugat, Martin and Leugering, Günter and Martin,
Alexander and Schmidt, Martin and Sirvent, Mathias and Wintergerst, David}, doi = {10.1007/s10589-017-9970-1}, faupublication = {yes}, journal = {Computational Optimization and Applications},
keywords = {Mixed-integer optimal control, Instantaneous control, Partial differential equations on graphs, Gas networks, Mixed-integer linear optimization.}, pages = {267-294}, peerreviewed = {Yes},
title = {{MIP}-based instantaneous control of mixed-integer {PDE}-constrained gas transport problems}, volume = {70}, year = {2018} } @misc{faucris.115699584, author = {Geißler, Björn and Kolb,
Oliver and Lang, Jens and Leugering, Günter and Martin, Alexander and Morsi, Antonio}, faupublication = {yes}, peerreviewed = {automatic}, title = {{Mixed} {Integer} {Linear} {Models} for the
{Optimization} of {Dynamical} {Transport} {Networks}}, year = {2010} } @article{faucris.124196424, abstract = {We introduce a mixed integer linear modeling approach for the optimization of dynamic
transport networks based on the piecewise linearization of nonlinear constraints and we show how to apply this method by two examples, transient gas and water supply network optimization. We state
the mixed integer linear programs for both cases and provide numerical evidence for their suitability.}, author = {Geißler, Björn and Kolb, Oliver and Lang, Jens and Leugering, Günter and Martin,
Alexander and Morsi, Antonio}, doi = {10.1007/s00186-011-0354-5}, faupublication = {yes}, journal = {Mathematical Methods of Operations Research}, keywords = {Mixed integer linear programming;
Piecewise linear approximation;Gas network optimization;Water network optimization}, pages = {339-362}, peerreviewed = {Yes}, title = {{Mixed} integer linear models for the optimization of dynamical
transport networks}, volume = {73}, year = {2011} } @article{faucris.123264724, abstract = {We consider hot forming processes, in which a metal solid body is deformed by several rolls in order to
obtain a desired final shape. To minimize cutting scrap and to ensure that this shape satisfies the required tolerances as precisely as possible, we formulate an optimal control problem where we use
the trajectories of the rolls as control functions. The deformation of the solid body is described through the basic equations of nonlinear continuum mechanics, which are here coupled with an
elasto-viscoplastic material model based on a multiplicative split of the deformation gradient. We assume that the deformations of the rolls can be neglected, thus we add unilateral frictional
contact boundary conditions, resulting in an evolutionary quasi-variational inclusion. The associated control-to-observation map is non-differentiable due to changes of state between elastic and
plastic material behavior, contact and separation and stick and slip motion, yet we still want to apply gradient-based methods to solve the optimal control problem and therefore have to make sure
that derivatives of cost functional and constraints exist. To resolve this issue, we first regularize all non-differentiabilities and subsequently apply the direct differentiation method to obtain
sensitivity information. Finally, we formulate a suitable algorithm and discuss numerical results for a real-world example to illustrate its capability. (C) 2017 Elsevier B.V. All rights reserved.},
author = {Werner, Stefan and Stingl, Michael and Leugering, Günter}, doi = {10.1016/j.cma.2017.03.006}, faupublication = {yes}, journal = {Computer Methods in Applied Mechanics and Engineering},
keywords = {Optimal control; Regularization; Sensitivity analysis; Nonlinear continuum mechanics; Frictional contact; Elasto-viscoplasticity}, pages = {442-471}, peerreviewed = {Yes}, title =
{{Model}-based control of dynamic frictional contact problems using the example of hot rolling}, volume = {319}, year = {2017} } @article{faucris.117817084, abstract = {Mathematical modeling of
biochemical pathways is an important resource in Synthetic Biology, as the predictive power of simulating synthetic pathways represents an important step in the design of synthetic metabolons. In
this paper, we are concerned with the mathematical modeling, simulation, and optimization of metabolic processes in biochemical microreactors able to carry out enzymatic reactions and to exchange
metabolites with their surrounding medium. The results of the reported modeling approach are incorporated in the design of the first microreactor prototypes that are under construction. These
microreactors consist of compartments separated by membranes carrying specific transporters for the input of substrates and export of products. Inside the compartments of the reactor multienzyme
complexes assembled on nano-beads by peptide adapters are used to carry out metabolic reactions. The spatially resolved mathematical model describing the ongoing processes consists of a system of
diffusion equations together with boundary and initial conditions. The boundary conditions model the exchange of metabolites with the neighboring compartments and the reactions at the surface of the
nano-beads carrying the multienzyme complexes. Efficient and accurate approaches for numerical simulation of the mathematical model and for optimal design of the microreactor are developed. As a
proof-of-concept scenario, a synthetic pathway for the conversion of sucrose to glucose-6-phosphate (G6P) was chosen. In this context, the mathematical model is employed to compute the
spatio-temporal distributions of the metabolite concentrations, as well as application relevant quantities like the outflow rate of G6P. These computations are performed for different scenarios,
where the number of beads as well as their loading capacity are varied. The computed metabolite distributions show spatial patterns, which differ for different experimental arrangements. Furthermore,
the total output of G6P increases for scenarios where microcompartimentation of enzymes occurs. These results show that spatially resolved models are needed in the description of the conversion
processes. Finally, the enzyme stoichiometry on the nano-beads is determined, which maximizes the production of glucose-6-phosphat}, author = {Elbinger, Tobias and Gahn, Markus and Hante, Falk and
Voll, Lars and Leugering, Günter and Knabner, Peter and Neuss-Radu, Maria}, doi = {10.3389/fbioe.2016.00013}, faupublication = {yes}, journal = {Frontiers in Bioengineering and Biotechnology},
peerreviewed = {Yes}, title = {{Model}-{Based} {Design} of {Biochemical} {Microreactors}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr{\_}2016/2016{\_}
ElbingerGahnNeusRaduHanteVollLeugering{\_}ModelBasedDesignOfBioMicroReact}, year = {2016} } @article{faucris.237823264, abstract = {In order to obtain high-quality particulate products with tailored
properties, process conditions and their evolution in time must be chosen appropriately. Although the efficiency of these products depends on their dispersity in several dimensions, in established
processes the particle size is usually the decisive variable to adjust. As part of the synthesis of these products, feedback modules are often incorporated so that a time-dependent ratio of the
obtained product can flow back into the system. Moreover, the synthesis should be an energy- and resource-efficient process. To provide a means of ensuring this requirement, a model- and
gradient-based, numerically efficient optimization tool for particle synthesis is presented which was developed to describe population balance equations incorporating feedback terms.}, author =
{Spinola, Michele and Keimer, Alexander and Segets, Doris and Leugering, Günter and Pflug, Lukas}, doi = {10.1002/ceat.201900515}, faupublication = {yes}, journal = {Chemical Engineering &
Technology}, keywords = {Feedback modules; Gradient-based optimization; Population balance equation; Ripening process}, note = {CRIS-Team Scopus Importer:2020-04-28}, pages = {896-903}, peerreviewed
= {Yes}, title = {{Model}-{Based} {Optimization} of {Ripening} {Processes} with {Feedback} {Modules}}, volume = {43}, year = {2020} } @incollection{faucris.241184099, abstract = {We consider the
processes of particle nucleation, growth, precipitation and ripening via modeling by nonlinear 1-D hyperbolic partial integro differential equations. The goal of this contribution is to provide a
concise predictive forward modeling of the processes including appropriate goal functions and to establish a mathematical theory for the open-loop optimization in this context. Beyond deriving
optimality conditions in the synthesis process, we present the application of a fully implicit method for Ostwald ripening of ZnO quantum dots which preserves its numeric stability even with respect
to the inherent high sensitivities and wide disparity of scales. FIMOR represents an appropriate method that can be integrated to subordinate optimization studies which enables its future application
in the context of continuous particle syntheses and microreaction technology (MRT}, author = {Gröschel, Michael and Peukert, Wolfgang and Leugering, Günter}, booktitle = {Trends in PDE Constrained
Optimization}, doi = {10.1007/978-3-319-05083-6{\_}30}, editor = {Günter Leugering; Peter Benner; Sebastian Engell; Andreas Griewank; Helmut Harbrecht; Michael Hinze; Rolf Rannacher; Stefan Ulbrich},
faupublication = {yes}, keywords = {Numerical simulation; Nonlinear integro-partial differential equation; Ostwald ripening; Optimization; Particle synthesis; Particle growth}, pages = {471-486},
peerreviewed = {unknown}, publisher = {Springer}, series = {International Series of Numerical Mathematics}, title = {{Modeling}, analysis and optimization of particle growth, nucleation and ripening
by the way of nonlinear hyperbolic integro-partial differential equations}, volume = {165}, year = {2014} } @article{faucris.112038344, abstract = {We consider networked transport systems defined on
directed graphs: the dynamics on the edges correspond to solutions of transport equations with space dimension one. In addition to the graph setting, a major consideration is the introduction and
propagation of discontinuities in the solutions when the system may discontinuously switch modes, naturally or as a hybrid control. This kind of switching has been extensively studied for ordinary
differential equations, but not much so far for systems governed by partial differential equations. In particular, we give well-posedness results for switching as a control, both in finite horizon
open loop operation and as feedback based on sensor measurements in the system. © 2008 Springer Science+Business Media, LLC.}, author = {Hante, Falk and Leugering, Günter and Seidman, Thomas I.}, doi
= {10.1007/s00245-008-9057-6}, faupublication = {yes}, journal = {Applied Mathematics and Optimization}, keywords = {Automated switching; Hybrid system; Networked system; Optimal control; Transport
equation}, note = {UnivIS-Import:2015-03-09:Pub.2009.nat.dma.lama1.modeli}, pages = {275-292}, peerreviewed = {Yes}, title = {{Modeling} and {Analysis} of {Modal} {Switching} in {Networked}
{Transport} {Systems}}, url = {http://www.springerlink.com/content/v31p353l09057015/}, volume = {59}, year = {2009} } @incollection{faucris.122683924, abstract = {The purpose of this paper is to
disduss, in a fairly heuristic way, certain aspects of modelling and controllability of flexible, multi-beam structures from the point of view of local modelling of the structure by distributed
parameter systems.}, address = {Berlin, Heidelberg}, author = {Langnese, John E. and Leugering, Günter and Schmidt, E. J. P. Georg}, booktitle = {System Modelling and Optimization - Proceedings of
the 15th IFIP Conference Zurich, Switzerland, September 2–6, 1991}, doi = {10.1007/BFb0113314}, editor = {L.D. Davisson; A.G.J. MacFarlane; H. Kwakernaak; J.L. Massey; Ya Z. Tsypkin; A.J. Viterbi;
Peter Kall}, faupublication = {no}, isbn = {978-3-540-47220-9}, month = {Jan}, pages = {467-480}, peerreviewed = {unknown}, publisher = {New York; Springer; 1999}, series = {Lecture Notes in Control
and Information Sciences}, title = {{MODELING} {AND} {CONTROLLABILITY} {OF} {NETWORKS} {OF} {THIN} {BEAMS}}, url = {http://link.springer.com/chapter/10.1007/BFb0113314}, volume = {180}, year = {1992}
} @article{faucris.121613404, abstract = {The purpose of this paper is to derive junction conditions for networks of thin elastic plates and to analyse the dynamic equations of such networks.
Junction conditions for networks of Kirchhoff plates and networks of Reissner-Mindlin plates are derived based on geometric considerations of the deformation at a junction. It is proved that the
dynamic system which describes the Reissner-Mindlin network is well-posed is an appropriate energy space. It is further established that the Kirchhoff network is obtained in the limit of the
Reissner-Mindlin network as the shear moduli go to infinity.}, author = {Langnese, John E. and Leugering, Günter}, doi = {10.1002/mma.1670160602}, faupublication = {no}, journal = {Mathematical
Methods in the Applied Sciences}, pages = {379-407}, peerreviewed = {Yes}, title = {{MODELING} {OF} {DYNAMIC} {NETWORKS} {OF} {THIN} {ELASTIC} {PLATES}}, url = {http://onlinelibrary.wiley.com/doi/
10.1002/mma.1670160602/full}, volume = {16}, year = {1993} } @article{faucris.117908824, abstract = {We derive a distributed-parameter model of a thin non-linear thermoelastic beam in three
dimensions. The beam can also be initially curved and twisted. Our main task is to formulate the non-homogeneous initial, boundary and node value problem associated with the dynamics of a network of
a finite number of such beams. The emphasis here is on a distributed-parameter modelling of the geometric and kinematic node conditions. The forces and couples appearing in the boundary and node
conditions can then be viewed as control variables. The analysis of the resulting control systems and their controllability and stabilizability properties is the subject of [25] and of forthcoming
papers.}, author = {Langnese, John E. and Leugering, Günter and Schmidt, E. J. P. Georg}, doi = {10.1002/mma.1670160503}, faupublication = {no}, journal = {Mathematical Methods in the Applied
Sciences}, pages = {327-358}, peerreviewed = {Yes}, title = {{MODELING} {OF} {DYNAMIC} {NETWORKS} {OF} {THIN} {THERMOELASTIC} {BEAMS}}, url = {http://onlinelibrary.wiley.com/doi/10.1002/
mma.1670160503/full}, volume = {16}, year = {1993} } @incollection{faucris.122007864, abstract = {This chapter focuses on acoustic, electromagnetic, elastic and piezo-electric wave propagation
through heterogenous layers. The motivation is provided by the demand for a better understanding of meta-materials and their possible construction. We stress the analogies between the mathematical
treatment of phononic, photonic and elastic meta-materials. Moreover, we treat the cloaking problem in more detail from an analytical and simulation oriented point of view. The novelty in the
approach presented here is with the interlinked homogenization- and optimization procedur}, author = {Leugering, Günter and Rohan, Eduard and Seifrt, Franti ek}, booktitle = {Wave Propagation in
Periodic Media}, doi = {10.2174/978160805150211001010197}, editor = {Matthias Ehrhardt}, faupublication = {yes}, note = {EAM Import::2019-03-12}, pages = {197-226}, peerreviewed = {unknown},
publisher = {Bentham Science}, series = {Progress in Computational Physics}, title = {{Modeling} of {Metamaterials} in {Wave} {Propagation}}, volume = {1}, year = {2010} } @incollection
{faucris.240105087, author = {Spinola, Michele and Keimer, Alexander and Segets, Doris and Pflug, Lukas and Leugering, Günter}, booktitle = {Dynamic Flowsheet Simulation of Solids Processes}, doi =
{10.1007/978-3-030-45168-4}, editor = {Heinrich, Stefan}, faupublication = {yes}, isbn = {978-3-030-45167-7}, pages = {549-578}, peerreviewed = {unknown}, publisher = {Springer International
Publishing}, title = {{Modeling}, {Simulation} and {Optimization} of {Process} {Chains}}, year = {2020} } @book{faucris.122826924, address = {Bayreuth}, author = {Rathmann, Wigand}, faupublication =
{no}, peerreviewed = {Yes}, publisher = {University of Bayreuth}, series = {Bayreuther Mathematische Schriften}, title = {{Modellierung}, {Simulation} und {Steuerung} von {Netzwerken} schwingender
{Balken} mittels dynamischer {Bereichszerlegung}}, volume = {60}, year = {2000} } @incollection{faucris.122007644, address = {Berlin}, author = {Gugat, Martin and Leugering, Günter and Schittkowski,
Klaus and Schmidt, E. J. P. Georg}, booktitle = {Online Optimization of Large Scale Systems}, doi = {10.1007/978-3-662-04331-8{\_}16}, faupublication = {no}, isbn = {3-540-42459-8}, note =
{UnivIS-Import:2015-04-20:Pub.2001.nat.dma.zentr.modell}, pages = {251-270}, peerreviewed = {unknown}, publisher = {Springer}, title = {{Modelling}, stabilization and control of flow in networks of
open channels}, year = {2001} } @article{faucris.107336064, abstract = {The production of nanoscaled particulate products with exactly pre-defined characteristics is of enormous economic relevance.
Although there are different particle formation routes they may all be described by one class of equations. Therefore, simulating such processes comprises the solution of nonlinear, hyperbolic
integro-partial differential equations. In our project we aim to study this class of equations in order to develop efficient tools for the identification of optimal process conditions to achieve
desired product properties. This objective is approached by a joint effort of the mathematics and the engineering faculty. Two model-processes are chosen for this study, namely a precipitation
process and an innovative aerosol process allowing for a precise control of residence time and temperature. Since the overall problem is far too complex to be solved directly a hierarchical sequence
of simplified problems has been derived which are solved consecutively. In particular, the simulation results are finally subject to comparison with experiments.}, author = {Gröschel, Michael and
Leugering, Günter and Peukert, Wolfgang}, doi = {10.1007/978-3-0348-0133-1{\_}28}, faupublication = {yes}, journal = {International Series of Numerical Mathematics}, keywords = {Population balance
equations; optimal control; model reduction; parameter identification}, pages = {541-559}, peerreviewed = {Yes}, title = {{Model} {Reduction}, {Structure}-property {Relations} and {Optimization}
{Techniques} for the {Production} of {Nanoscale} {Particles}}, url = {http://link.springer.com/chapter/10.1007%2F978-3-0348-0133-1{\_}28}, year = {2011} } @article{faucris.122009624, abstract = {We
present a mathematical framework for the so-called multidisciplinary free material optimization (MDFMO) problems, a branch of structural optimization in which the full material tensor is considered
as a design variable. We extend the original problem statement by a class of generic constraints depending either on the design or on the state variables. Among the examples are local stress or
displacement constraints. We show the existence of optimal solutions for this generalized free material optimization (FMO) problem and discuss convergent approximation schemes based on the finite
element method.}, author = {Haslinger, J. and Kocvara, Michal and Leugering, Günter and Stingl, Michael}, doi = {10.1137/090774446}, faupublication = {yes}, journal = {SIAM Journal on Applied
Mathematics}, pages = {2709}, peerreviewed = {Yes}, title = {{Multidisciplinary} {Free} {Material} {Optimization}}, volume = {70}, year = {2010} } @article{faucris.111527724, author = {Jahn, Johannes
and Hillermeier, C.}, faupublication = {yes}, journal = {Surveys on Mathematics for Industry}, note = {UnivIS-Import:2015-03-09:Pub.2005.nat.dma.pama21.multio}, pages = {1-42}, peerreviewed =
{unknown}, title = {{Multiobjective} optimization: survey of methods and industrial applications}, volume = {11}, year = {2005} } @article{faucris.111690964, abstract = {This paper presents a
multiobjective search algorithm with subdivision technique (MOSAST) for the global solution of multiobjective constrained optimization problems with possibly noncontinuous objective or constraint
functions. This method is based on a random search method and a new version of the Graef-Younes algorithm and it uses a subdivision technique. Numerical results are given for bicriterial test
problems. © 2006 Springer Science + Business Media, LLC.}, author = {Jahn, Johannes}, doi = {10.1007/s10589-006-6450-4}, faupublication = {yes}, journal = {Computational Optimization and
Applications}, keywords = {Global optimization; Random search; Vector optimization}, note = {UnivIS-Import:2015-03-09:Pub.2006.nat.dma.pama21.multio}, pages = {161-175}, peerreviewed = {Yes}, title =
{{Multiobjective} {Search} {Algorithm} with {Subdivision} {Technique}}, volume = {35}, year = {2006} } @article{faucris.266498134, abstract = {In this work, we are interested in tree-shaped networks
of freely vibrating beams which are geometrically exact (GEB) - in the sense that large motions (deflections, rotations) are accounted for in addition to shearing - and linked by rigid joints. For
the intrinsic GEB formulation, namely that in terms of velocities and internal forces/moments, we derive transmission conditions and show that the network is locally in time well-posed in the
classical sense. Applying velocity feedback controls at the external nodes of a star-shaped network, we show by means of a quadratic Lyapunov functional and the theory developed by Bastin & Coron in
[2] that the zero steady state of this network is exponentially stable for the H-1 and H-2 norms. The major obstacles to overcome in the intrinsic formulation of the GEB network, are that the
governing equations are semilinar, containing a quadratic nonlinearity, and that linear lower order terms cannot be neglected.}, author = {Rodriguez, Charlotte}, doi = {10.3934/mcrf.2021002},
faupublication = {yes}, journal = {Mathematical Control and Related Fields}, note = {CRIS-Team WoS Importer:2021-11-26}, peerreviewed = {Yes}, title = {{NETWORKS} {OF} {GEOMETRICALLY} {EXACT}
{BEAMS}: {WELL}-{POSEDNESS} {AND} {STABILIZATION}}, year = {2020} } @incollection{faucris.121574244, address = {Berlin}, author = {Jahn, Johannes}, booktitle = {Operations Research Proceedings 1983},
editor = {H. Steckhan, W. Bühler, K.-E. Jäger, Ch. Schneeweiß, J. Schwarze}, faupublication = {no}, note = {UnivIS-Import:2015-04-17:Pub.1984.nat.dma.pama21.neuere}, pages = {511-519}, peerreviewed =
{unknown}, publisher = {Springer}, title = {{Neuere} {Entwicklungen} in der {Vektoroptimierung}}, year = {1984} } @article{faucris.110289344, abstract = {For a system that is governed by the
isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order
quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a
finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict H2-Lyapunov function and show that the boundary feedback constant can be chosen such that the
H2-Lyapunov function and hence also the H2-norm of the difference between the non-stationary and the stationary state decays exponentially with tim}, author = {Gugat, Martin and Leugering, Günter and
Wang, Ke}, doi = {10.3934/mcrf.2017015}, faupublication = {yes}, journal = {Mathematical Control and Related Fields}, keywords = {Boundary feedback control, feedback stabilization, exponential
stability, isothermal Euler equations, second-order quasilinear equation, Lyapunov function, stationary state, non-stationary state, gas pipeline}, pages = {419 - 448}, peerreviewed = {Yes}, title =
{{Neumann} boundary feedback stabilization for a nonlinear wave equation: {A} strict {H2}-{Lyapunov} function}, volume = {7}, year = {2017} } @article{faucris.107392824, abstract = {For a system that
is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a
second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary
state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict H-2-Lyapunov function and show that the boundary feedback constant can be chosen such
that the H-2-Lyapunov function and hence also the H-2-norm of the difference between the non-stationary and the stationary state decays exponentially with time.}, author = {Leugering, Günter and
Gugat, Martin and Wang, Ke}, doi = {10.3934/mcrf.2017015}, faupublication = {yes}, journal = {Mathematical Control and Related Fields}, keywords = {Boundary feedback control;feedback stabilization;
exponential stability;isothermal Euler equations;second-order quasilinear equation;Lyapunov function;stationary state;non-stationary state;gas pipeline}, pages = {419-448}, peerreviewed = {Yes},
title = {{NEUMANN} {BOUNDARY} {FEEDBACK} {STABILIZATION} {FOR} {A} {NONLINEAR} {WAVE} {EQUATION}: {A} {STRICT} {H}-2-{LYAPUNOV} {FUNCTION}}, volume = {7}, year = {2017} } @article{faucris.112460964,
abstract = {In this paper we study a set optimization problem (SOP), i. e. we minimize a set-valued objective map F, which takes values on a real linear space Y equipped with a pre-order induced by a
convex cone K. We introduce new order relations on the power set P(Y) of Y (or on a subset of it), which are more suitable from a practical point of view than the often used minimizers in set
optimization. Next, we propose a simple two-steps unifying approach to studying (SOP) w. r. t. various order relations. Firstly, we extend in a unified scheme some basic concepts of vector
optimization, which are defined on the space Y up to an arbitrary nonempty pre-ordered set(Q) without any topological or linear structure. Namely, we define the following concepts w. r. t. the
pre-order : minimal elements, semicompactness, completeness, domination property of a subset of Q, and semicontinuity of a set-valued map with values in Q in a topological setting. Secondly, we
establish existence results for optimal solutions of (SOP), when F takes values on(Q) from which one can easily derive similar results for the case, when F takes values on P(Y) equipped with various
order relations. © 2010 Springer Science+Business Media, LLC.}, author = {Jahn, Johannes and Ha, T.X.D.}, doi = {10.1007/s10957-010-9752-8}, faupublication = {yes}, journal = {Journal of Optimization
Theory and Applications}, keywords = {Existence results; Order relations; Set optimization}, note = {UnivIS-Import:2015-03-09:Pub.2011.nat.dma.pama21.neword}, pages = {209-236}, peerreviewed = {Yes},
title = {{New} {Order} {Relations} in {Set} {Optimization}}, volume = {148}, year = {2011} } @article{faucris.119850544, author = {Hante, Falk and Mommer, Mario and Potschka, Andreas}, doi = {10.1137
/140967969}, faupublication = {yes}, journal = {SIAM Journal on Numerical Analysis}, pages = {2206-2225}, peerreviewed = {Yes}, title = {{Newton}-{Picard} preconditioners for time-periodic parabolic
optimal control problems}, volume = {53}, year = {2015} } @book{faucris.119359504, author = {Pratelli, Aldo and Leugering, Günter}, doi = {10.1007/978-3-319-17563-8}, faupublication = {yes}, isbn =
{978-3-319-17563-8}, month = {Jan}, pages = {VII-VIII}, peerreviewed = {unknown}, publisher = {Springer International Publishing}, series = {International Series of Numerical Mathematics}, title =
{{New} {Trends} in {Shape} {Optimization}}, url = {http://link.springer.com/book/10.1007/978-3-319-17563-8}, volume = {166}, year = {2015} } @incollection{faucris.116884504, abstract = {We consider a
network where a dynamical process governed by hyperbolic conservation laws takes place. At the vertices, the conservation laws are coupled by algebraic node conditions. The system is also controlled
at the vertices through these conditions. To solve problems of optimal control for such a system, an adjoint sensitivity calculus is useful since it allows the efficient evaluation of the gradient of
the objective function. We present such a calculus in the framework of classical solutions.}, address = {Boca Raton}, author = {Gugat, Martin}, booktitle = {Control and Boundary Analysis}, editor =
{John Cagnol and Jean-Paul Zolesio}, faupublication = {yes}, keywords = {Sensitivy calculus, adkoint system}, note = {UnivIS-Import:2015-04-20:Pub.2005.nat.dma.lama1.nodalc}, pages = {201-214},
peerreviewed = {Yes}, publisher = {Chapman & Hall/CRC Press}, series = {Pure and Applied Mathematics}, title = {{Nodal} {Control} of {Conservation} {Laws} on {Networks}}, volume = {240}, year =
{2005} } @article{faucris.262677645, abstract = {In this work, we consider networks of so-called geometrically exact beams, namely, shearable beams that may undergo large motions. The corresponding
mathematical model, commonly written in terms of displacements and rotations expressed in a fixed basis (Geometrically Exact Beam model, or GEB), has a quasilinear governing system. However, the
model may also be written in terms of intrinsic variables expressed in a moving basis attached to the beam (Intrinsic GEB model, or IGEB) and while the number of equations is then doubled, the latter
model has the advantage of being of first-order, hyperbolic and only semilinear. First, for any network, we show the existence and uniqueness of semi-global in time classical solutions to the IGEB
model (i.e., for arbitrarily large time intervals, provided that the data are small enough). Then, for a specific network containing a cycle, we address the problem of local exact controllability of
nodal profiles for the IGEB model – we steer the solution to satisfy given profiles at one of the multiple nodes by means of controls applied at the simple nodes – by using the constructive method of
Zhuang, Leugering and Li (2018) [52]. Afterwards, for any network, we show that the existence of a unique classical solution to the IGEB network implies the same for the corresponding GEB network, by
using that these two models are related by a nonlinear transformation. In particular, this allows us to give corresponding existence, uniqueness and controllability results for the GEB network.},
author = {Leugering, Günter and Rodriguez, Charlotte and Wang, Yue}, doi = {10.1016/j.matpur.2021.07.007}, faupublication = {yes}, journal = {Journal De Mathematiques Pures Et Appliquees}, keywords =
{Geometrically exact beam; Networks; Nodal profile controllability; Well-posedness}, note = {CRIS-Team Scopus Importer:2021-08-13}, peerreviewed = {Yes}, title = {{Nodal} profile control for networks
of geometrically exact beams}, year = {2021} } @inproceedings{faucris.234141280, author = {Keimer, Alexander and Pflug, Lukas and Spinola, Michele}, booktitle = {10}, date = {2018-06-25/2018-06-29},
editor = {AIMS Series on Applied Mathematics}, faupublication = {yes}, pages = {475-482}, peerreviewed = {unknown}, title = {{Nonlocal} balance laws - results on existence, uniqueness and
regularity}, url = {https://www.aimsciences.org/book/AM/volume/Volume 10}, venue = {Pennsylvania State University, University Park (USA)}, year = {2020} } @article{faucris.228833204, abstract = {This
study considers nonlocal conservation laws in which the velocity depends nonlocally on the solution not in real time but in a time-delayed manner. Nonlocal refers to the fact that the velocity of the
conservation law depends on the solution integrated over a specific area in space. In every model modelling human’s behavior a time delay as reaction/response time is crucial. We distinguish a so
called nonlocal classical delay model where only the velocity of the conservation law is delayed from a more realistic nonlocal delay model where also a shift backwards in space is considered. For
both models we show existence and uniqueness of the solutions and study their analytical properties. We also present a direct application in traffic flow modelling. We also show that for delay
approaching zero, the solutions of the considered delayed models converge to the solutions of the non-delayed models in the proper topology. Finally, a comprehensive numerical study illustrating
exemplary the impacts of delay and nonlocality are presented and compared with nonlocal models without delay, as well as the corresponding local models.}, author = {Keimer, Alexander and Pflug,
Lukas}, doi = {10.1007/s00030-019-0597-z}, faupublication = {yes}, journal = {Nodea-Nonlinear Differential Equations and Applications}, keywords = {Convergence for vanishing delay; Delay; Existence;
LWR PDE with delay; Nonlocal balance laws; Nonlocal conservation laws; Traffic flow modelling with delay; Uniqueness}, note = {CRIS-Team Scopus Importer:2019-11-08}, peerreviewed = {Yes}, title =
{{Nonlocal} conservation laws with time delay}, volume = {26}, year = {2019} } @article{faucris.216448317, abstract = {We consider a nonlocal conservation law on a bounded spatial domain and show
existence and uniqueness of weak solutions for nonnegative flux function and left-hand-side boundary datum. The nonlocal term is located in the flux function of the conservation law, averaging the
solution by means of an integral at every spatial coordinate and every time, forward in space. This necessitates the prescription of a kind of right-hand-side boundary datum, the external impact on
the outflow. The uniqueness of the weak solution follows without prescribing an entropy condition. Allowing the velocity to become zero (also dependent on the nonlocal impact) offers more realistic
modeling and significantly higher applicability. The model can thus be applied to traffic flow, as suggested for unbounded domains in [S. Blandin and P. Goatin, Numer. Math., 132 (2016), pp.
217-241], [P. Goatin and S. Scialanga, Netw. Hetereog. Media, 11 (2016), pp. 107-121]. It possesses finite acceleration and can be interpreted as a nonlocal approximation of the famous "local"
Lighthill-Whitham-Richards model [M. Lighthill and G. Whitham, Proc. Roy. Soc. London Ser. A, 229 (1955), pp. 281-316], [P. I. Richards, Oper. Res., 4 (1956), pp. 42-51]. Several numerical examples
are presented and discussed also with respect to the reasonableness of the required assumptions and the model itself.}, author = {Pflug, Lukas and Spinola, Michele and Keimer, Alexander}, doi =
{10.1137/18M119817X}, faupublication = {yes}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {first order macroscopic traffic flow model with finite acceleration;nonlocal conservation
law;initial boundary value problem;method of characteristics;fixed-point problem}, month = {Jan}, pages = {6271-6306}, peerreviewed = {Yes}, title = {{NONLOCAL} {SCALAR} {CONSERVATION} {LAWS} {ON}
{BOUNDED} {DOMAINS} {AND} {APPLICATIONS} {IN} {TRAFFIC} {FLOW}}, volume = {50}, year = {2018} } @article{faucris.246705191, abstract = {In this contribution, we study the existence and uniqueness of
nonlocal transport equations. The term "nonlocal" refers to the fact that the flux function's derivative will be integrated over a neighborhood of the corresponding space-time coordinate. We will
demonstrate existence and uniqueness of weak solutions for TV ∩ L∞ initial datum and provide stability estimates. Moreover, we investigate the convergence of the nonlocal transport equation to the
corresponding local conservation law when the nonlocal reach tends to zero. For quadratic flux functions (including Burgers' equation and the Lighthill-Whitham-Richards traffic flow model), we
establish convergence to a weak solution of the local conservation law for "symmetric" nonlocal terms. For specific quasi-convex and quasiconcave initial datum we even obtain convergence to the local
entropy solution. We demonstrate that for "nonsymmetric" nonlocal approximations the solution cannot converge to the entropy solution or even a weak solution. We conclude with additional numerical
examples showing that convergence appears to hold for more general initial datum.}, author = {Coron, Jean Michel and Keimer, Alexander and Pflug, Lukas}, doi = {10.1137/20M1331652}, faupublication =
{yes}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {convergence nonlocal to local; entropy solution; nonlocal advection equation; nonlocal balance laws; nonlocal conservation laws;
nonlocal transport equations}, note = {CRIS-Team Scopus Importer:2020-12-11}, pages = {5500-5532}, peerreviewed = {Yes}, title = {{Nonlocal} {Transport} {Equations}-{Existence} and {Uniqueness} of
{Solutions} and {Relation} to the {Corresponding} {Conservation} {Laws}}, volume = {52}, year = {2020} } @incollection{faucris.289687828, abstract = {We consider nonoverlapping domain decomposition
methods for p-type ordinary and partial differential equations and corresponding optimal control problems on metric graphs. As an exemplary context, we choose a doubly nonlinear p-parabolic model
with [Formula presented] which can be retrieved from the one-dimensional Euler system and that has come to be known as friction dominated flow in gas pipe networks. We introduce an optimal control
problem for such systems on a metric graph, where the pipes are represented by the edges. We then apply the rolling horizon approach and obtain a sequence of static optimal control problems of
p-type. The main interest of this survey article is to describe and utilize nonoverlapping domain decomposition procedures for such optimal control problems in the context of virtual controls which
lead to a decomposition of the entire optimality system such that the decomposed system is itself the optimality system of an optimal control problem. This transfers the parallel iterative domain
decomposition method into a sequence of parallel optimal control problems. We depart from the classical domain decomposition methods described by P.L. Lions (1990) and J.L. Lions and O. Pironneau
(1998, 1999, 2000) and extend those to problems on metric graphs. We provide some numerical simulations.}, author = {Leugering, Günter}, booktitle = {Handbook of Numerical Analysis}, doi = {10.1016/
bs.hna.2022.11.002}, editor = {Emmanuel Trélat, Enrique Zuazua}, faupublication = {yes}, keywords = {Domain decomposition; Instantaneous control; Optimal control; Optimality system; p-Laplace problem
on a graph; p-parabolic problems; Pdes on graphs}, month = {Jan}, note = {CRIS-Team Scopus Importer:2023-02-24}, pages = {217-260}, peerreviewed = {unknown}, publisher = {Elsevier B.V.}, title =
{{Nonoverlapping} domain decomposition and virtual controls for optimal control problems of p-type on metric graphs}, volume = {24}, year = {2023} } @article{faucris.320535669, abstract = {We
consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional
time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain
decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is
iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint
states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on
the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as
first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative,
along with two examples in order to verify the method.}, author = {Leugering, Günter and Mehandiratta, Vaibhav and Mehra, Mani}, doi = {10.3390/fractalfract8030129}, faupublication = {yes}, journal =
{Fractal and Fractional}, keywords = {Caputo fractional derivative; domain decomposition; optimal control; time-fractional diffusion equation}, note = {CRIS-Team Scopus Importer:2024-04-05},
peerreviewed = {Yes}, title = {{Non}-{Overlapping} {Domain} {Decomposition} for {1D} {Optimal} {Control} {Problems} {Governed} by {Time}-{Fractional} {Diffusion} {Equations} on {Coupled} {Domains}:
{Optimality} {System} and {Virtual} {Controls}}, volume = {8}, year = {2024} } @article{faucris.119464224, abstract = {We consider optimal control problems for gas flow in pipeline networks. The
equations of motion are taken to be represented by a first-order system of hyperbolic semilinear equations derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal
control problem on a network and introduce a tailored time discretization thereof. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper
is concerned with a nonoverlapping domain decomposition of the optimal control problem on the graph into local problems on smaller sub-graphs - ultimately on single edges. We prove convergence of the
domain decomposition method on networks and study the wellposedness of the corresponding time-discrete optimal control problems. The point of the paper is that we establish virtual control problems
on the decomposed subgraphs such that the corresponding optimality systems are in fact equal to the systems obtained via the domain decomposition of the entire optimality system.}, author =
{Leugering, Günter and Martin, Alexander and Schmidt, Martin and Sirvent, Mathias}, faupublication = {yes}, journal = {Control and Cybernetics}, keywords = {Optimal control; Gas networks; Euler's
equation; Semilinear PDE; Nonoverlapping domain decomposition}, pages = {191-225}, peerreviewed = {Yes}, title = {{Nonoverlapping} {Domain} {Decomposition} for {Optimal} {Control} {Problems} governed
by {Semilinear} {Models} for {Gas} {Flow} in {Networks}}, volume = {46}, year = {2017} } @article{faucris.113909884, abstract = {We consider the problem to control a vibrating string to rest in a
given finite time. The string is fixed at one end and controlled by Neumann boundary control at the other end. We give an explicit representation of the L ^2-norm minimal control in terms of the
given initial state. We show that if the initial state is sufficiently regular, the same control is also L ^ p -norm minimal for p > 2.}, author = {Gugat, Martin}, doi = {10.1007/s40065-014-0110-9},
faupublication = {yes}, journal = {Arabian Journal of Mathematics}, keywords = {35L05; 49J20; 93C20}, note = {UnivIS-Import:2015-03-09:Pub.2014.nat.dma.zentr.normmi}, pages = {41-58}, peerreviewed =
{unknown}, title = {{Norm}-minimal {Neumann} boundary control of the wave equation}, url = {http://link.springer.com/article/10.1007/s40065-014-0110-9}, volume = {4}, year = {2014} } @article
{faucris.219579007, author = {Ray, Nadja and Oberlander, Jens and Frolkovic, Peter}, doi = {10.1007/s10596-019-09876-x}, faupublication = {yes}, journal = {Computational Geosciences}, peerreviewed =
{Yes}, title = {{Numerical} investigation of a fully coupled micro-macro model for mineral dissolution and precipitation}, year = {2019} } @inproceedings{faucris.124257144, author = {Rathmann, Wigand
and Kaltenbacher, Manfred and Krug, Andreas and Landes, Hermann and Rausch, Martin and Dietz, Peter and Vogel, Frank}, booktitle = {23. CAD-FEM Users' Meeting}, faupublication = {no}, peerreviewed =
{No}, publisher = {CAD-FEM GmbH Grafing}, title = {{Numerical} {Optimization} of {Time}-dependent {Electromechanical} {Systems}}, year = {2005} } @inproceedings{faucris.113999424, author = {Rathmann,
Wigand and Vogel, Frank and Landes, Hermann}, booktitle = {22. CAD-FEM Users' Meeting}, faupublication = {no}, peerreviewed = {No}, publisher = {CAD-FEM GmbH Grafing}, title = {{Numerical}
{Optimization} of {Time}-dependent {Electromechanical} {Systems}}, year = {2004} } @masterthesis{faucris.118445184, abstract = {In der vorliegenden Arbeit werden zwei numerische Verfahren für die
Registrierung medizinischer Bilddaten vorgestellt. Beide Verfahren beruhen auf der numerischen Lösung eines Optimierungsproblems, welches eine Transportgleichung als Nebenbedingung besitzt. Der
Lösungsalgorithmus koppelt ein Innere-Punkte-Verfahren für das Optimierungsproblem mit einem Finite-Elemente-Verfahren für die Transportgleichung.}, author = {Hild, Johannes}, faupublication = {yes},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{Numerische} {Verfahren} für optimalen {Massentransport} in der {Registrierung} medizinischer {Bilddaten}}, year = {2007} }
@article{faucris.246376475, abstract = {A 1-parameter initial boundary value problem for the linear homogeneous degenerate wave equation utt(t, x; α)−(a(x; α) ux(t, x; α))x = 0 (JODEA, 27(2), 29 -
44), where: 1) (t, x) ∈ [0, T] ×[−l, +l]; 2) the weight function a(x; α): a) a0^xc^α, 0 6 |x| 6 c; b) a0, c 6 |x| 6 l; c) a0 is a constant reference value; and 3) the parameter α ∈ (0, +∞); is
considered. Using a string analogy, the IBVP can be treated as an attempt to set an initially fixed 'string' in motion, the left end of the 'string' being fixed, whereas the right end being forced to
move. It has been proved, using the methods of Frobenius and separation of variables, that: 1) there exist 6 series solutions u(t, x; α), (t, x) ∈ [0, T]×[−c, +c], of the degenerate wave equation; 2)
the only series solution, having continuous and continuously differentiable flux f(a, u) = −aux, reads u(t, x; α) = Uα0(t) + Uα1(t)|x|^θ + Uα2(t)|x|^2θ +..., where a) θ = 2 − α is a derived
parameter; b) the coefficient functions obey the following linear recurrence relations: Uα,µ^00−1(t) = µθ [(µ−1) θ + 1] c^−αa0 Uα,µ(t), µ∈N. It has been revealed that a nonlinear change of the
independent variables (t, x) → (τ, ξ) transforms: 1) the degenerate wave equation to the wave equation υττ − υξξ = ξρ, or rewritten as the balance law πτ + ϕξ = ρ, where π = υτ, −ϕ(υ; α) = υξ + ξρ, ρ
(υ; α) = θ ξ2 α υ having: a) no singularity in its principal part (due to inflation of the degeneracy), and b) the only series solution of the form υ(τ, ξ; α) = Vα0(τ) + Vα1(τ) ξ^2 + Vα2(τ) ξ^4 +...
(out of 5 existing and found similarly to those of the degenerate wave equation), leading to the continuous and continuously differentiable regularized flux ϕ(υ; α) and the continuous regularized
source term ρ(υ; α), where υ(τ, ξ; α) = υ(τ, ξ; α) − υ(τ, 0; α); 2) the IBVP for the degenerate wave equation to the IBVP for the transformed wave equation. It has been shown, that if α∈(0, 2): 1)
the above results are valid; 2) the state of being fixed for the 'string' is not necessary for (t, x) ∈ [0, T] ×[−l, 0], that is a traveling wave could pass the degeneracy and excite vibrations of
the 'string' between its fixed end and the point of degeneracy.}, author = {Borsch, Vladimir L. and Kogut, Peter I. and Leugering, Günter}, doi = {10.15421/142001}, faupublication = {yes}, journal =
{Journal of Optimization, Differential Equations and Their Applications}, keywords = {Conservation and balance laws; Degenerate wave equation; Exact solutions; Inflation of singularity;
Regularization of the flux; Separation of variables; Series solutions; The Bessel functions; The flux; The Frobenius method}, note = {CRIS-Team Scopus Importer:2020-12-04}, pages = {1-42},
peerreviewed = {Yes}, title = {{On} an initial boundary-value problem for {1D} hyperbolic equation with interior degeneracy: {Series} solutions with the continuously differentiable fluxes}, volume =
{28}, year = {2020} } @article{faucris.107368844, abstract = {We consider the in-plane motion of elastic strings on tree-like network, observed from the ‘leaves’. We investigate the inverse problem
of recovering not only the physical properties, i.e. the ‘optical lengths’ of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching
edges. To this end we use the Boundary Control method for wave equations on graphs established in [4, 7]. It is shown that under generic assumptions the inverse problem can be solved by applying
measurements at all leaves, the root of the tree being fixed.}, author = {Avdonin, Sergei and Leugering, Günter and Mikhaylov, Victor}, doi = {10.1002/zamm.200900295}, faupublication = {yes}, journal
= {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik}, pages = {136--150}, peerreviewed = {Yes}, title = {{On} an inverse problem for tree-like networks of elastic strings}, volume = {90},
year = {2010} } @article{faucris.216448800, author = {Pflug, Lukas and Keimer, Alexander}, doi = {10.1016/j.jmaa.2019.03.063}, faupublication = {yes}, journal = {Journal of Mathematical Analysis and
Applications}, keywords = {Convergence of the nonlocal model to corresponding local model; LWR PDE; Entropy solution; Traffic flow modelling; Nonlocal balance laws}, pages = {1927-1955}, peerreviewed
= {Yes}, title = {{On} approximation of local conservation laws by nonlocal conservation laws}, volume = {475}, year = {2019} } @article{faucris.119942944, abstract = {
A vibrating plate is here taken to satisfy the model equation:u[tt] + Δ^2u = 0 (whereΔ^2u:= Δ(Δu); Δ = Laplacian) with boundary conditions of the form:u[v] = 0 and(Δu)[v] = ϕ = control. Thus, the
state is the pair [u, u[t]] and controllability means existence ofϕ on Σ:= (0,T)×∂Ω transfering ‘any’[u, u[t]][0] to ‘any’[u, u[t]][T]. The formulation is given by eigenfunction expansion and
duality. The substantive results apply to a rectangular plate. For largeT one has such controllability with∥ϕ∥ = O(T^−1/2). More surprising is that (based on a harmonic analysis estimate [11]) one
has controllability for arbitrarily short times (in contrast to the wave equation:u[tt] = Δu) with log∥ϕ∥ = O(T^−1) asT→0. Some related results on minimum time control are also included.
}, author = {Leugering, Günter and Krabs, W. and Seidman, Thomas I.}, doi = {10.1007/BF01442208}, faupublication = {no}, journal = {Applied Mathematics and Optimization}, pages = {205-229},
peerreviewed = {Yes}, title = {{ON} {BOUNDARY} {CONTROLLABILITY} {OF} {A} {VIBRATING} {PLATE}}, url = {http://link.springer.com/article/10.1007/BF01442208}, volume = {13}, year = {1985} } @article
{faucris.117909924, abstract = {This paper is concerned with boundary control of one-dimensional vibrating media whose motion is governed by a wave equation with a 2n-order spatial self-adjoint and
positive-definite linear differential operator with respect to 2n boundary conditions. Control is applied to one of the boundary conditions and the control function is allowed to vary in the Sobolev
space W, ^p for p∈[2, ∞] With the aid of Banach space theory of trigonometric moment problems, necessary and sufficient conditions for null-controllability are derived and applied to vibrating
strings and Euler beams.
For vibrating strings also, null-controllability by L(p)-controls on the boundary is shown by a direct method which makes use of d'Alembert's solution formula for the wave equation.}, author =
{Krabs, W. and Leugering, Günter}, doi = {10.1002/mma.1670170202}, faupublication = {no}, journal = {Mathematical Methods in the Applied Sciences}, pages = {77-93}, peerreviewed = {Yes}, title =
{{ON} {BOUNDARY} {CONTROLLABILITY} {OF} {ONE}-{DIMENSIONAL} {VIBRATING} {SYSTEMS} {BY} {W}(0)1,{P}-{CONTROLS} {FOR} p∈[2, ∞]}, url = {http://onlinelibrary.wiley.com/doi/10.1002/mma.1670170202/
abstract}, volume = {17}, year = {1994} } @article{faucris.117910144, abstract = {It is shown that a general isotropic viscoelastic solid with non vanishing Newtonian viscosity is never exactly
controllable using L[2]-boundary controls. For some models it is known that even spectral controllability does not hold. Here we show, thereby extending results obtained in Leugering and Schmidt
[10], that the general model is approximatively controllable under some reasonable assumptions.}, author = {Leugering, Günter}, doi = {10.1007/BFb0002592}, faupublication = {no}, journal = {Lecture
Notes in Control and Information Sciences}, month = {Jan}, pages = {190-201}, peerreviewed = {Yes}, title = {{ON} {BOUNDARY} {CONTROLLABILITY} {OF} {VISCOELASTIC} {SYSTEMS} - {Proceedings} of the
{IFIP} {WG} 7.2 {Working} {Conference} {Santiago} de {Compostela}, {Spain}, {July} 6–9, 1987}, url = {http://link.springer.com/chapter/10.1007/BFb0002592}, volume = {114}, year = {1989} } @article
{faucris.117910364, author = {Leugering, Günter}, doi = {10.1007/BFb0041994}, faupublication = {no}, journal = {Lecture Notes in Control and Information Sciences}, month = {Jan}, pages = {234-252},
peerreviewed = {Yes}, title = {{ON} {BOUNDARY} {CONTROLLABILITY} {OF} {VOLTERRA} {INTEGRODIFFERENTIAL} {EQUATIONS} {IN} {HILBERT}-{SPACES}}, url = {http://link.springer.com/chapter/10.1007/
BFb0041994}, volume = {102}, year = {1987} } @article{faucris.264862056, abstract = {In this paper, we study exact boundary controllability for a linear wave equation with strong and weak interior
degeneration of the coefficient in the principle part of the elliptic operator. The objective is to provide a well-posedness analysis of the corresponding system and derive conditions for its
controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the
rate of degeneracy for both exact boundary controllability and the lack thereof.}, author = {Kogut, Peter I. and Kupenko, Olga P. and Leugering, Günter}, doi = {10.1002/mma.7811}, faupublication =
{yes}, journal = {Mathematical Methods in the Applied Sciences}, keywords = {boundary control; degenerate wave equation; exact controllability; existence result; weighted Sobolev spaces}, note =
{CRIS-Team Scopus Importer:2021-10-08}, peerreviewed = {Yes}, title = {{On} boundary exact controllability of one-dimensional wave equations with weak and strong interior degeneration}, year = {2021}
} @article{faucris.120846704, abstract = {It is shown that a cantilevered beam with weak viscoelastic damping of Boltzmann-type can be uniformly stabilised by velocity feedback applied as a shearing
force at the free end of the beam. Estimates for the viscoelastic energy are derived using the energy multiplier method. The energy decay is related to the decay of the relaxation modulus associated
with the viscoelastic material.}, author = {Leugering, Günter}, doi = {10.1017/S0308210500024264}, faupublication = {no}, journal = {Proceedings of the Royal Society of Edinburgh Section
A-Mathematics}, month = {Jan}, pages = {57-69}, peerreviewed = {Yes}, title = {{ON} {BOUNDARY} {FEEDBACK} {STABILISABILITY} {OF} {A} {VISCOELASTIC} {BEAM}}, url = {https://www.cambridge.org/core/
journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/on-boundary-feedback-stabilisability-of-a-viscoelastic-beam/F4A796392C45ECEEE3A0142E33FAA739}, volume = {114},
year = {1990} } @article{faucris.119952184, abstract = {We consider a tripod as an exemplaric network of strings. We know that such a network is exactly controllable in the natural finite energy
space, if, e.g., the simple nodes are controlled by Dirichlet controls in (HL)-L-1(0, T). Assume that we want to calculate the corresponding norm-minimal controls using semi-discretization in space.
We then obtain a system of coupled second-order-in-time ordinary differential equations with three control inputs. Controllability of the latter system can easily been checked by Kalman's rank
condition on each space discretization level h. One expects, as h tends to zero, that the exact controllability of the continuous system is revealed. This expectation is frustrated, as has been shown
by Infante and Zuazua (1998) for a. single string and by Zuazua (1999) for a membrane. Indeed, it was shown there that uniformity of observability estimates is lost in the limit. On the other hand,
spectral filtering allows to cure this pathology. We show in this paper that similar results hold for our string network. The generalization to arbitrary networks of strings in the out-of-the-plane
as well as in the in-plane or 3-d-setup is then a technical matter. Therefore, this paper essentially extends the existing results to semidiscretizations of wave equations on arbitrary irregular
computational grids.}, author = {Brauer, U. and Leugering, Günter}, faupublication = {no}, journal = {Control and Cybernetics}, keywords = {network of strings;semidiscretization;lack of uniform
observability;filtering}, month = {Jan}, pages = {421-447}, peerreviewed = {Yes}, title = {{On} boundary observability estimates for semi-discretizations of a dynamic network of elastic strings}, url
= {http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.baztech-article-BAT2-0001-1167}, volume = {28}, year = {1999} } @article{faucris.106690364, author = {Hante, Falk and Sigalotti, Mario and
Tucsnak, Marius}, doi = {10.1016/j.jde.2012.01.037}, faupublication = {no}, journal = {Journal of Differential Equations}, pages = {5569--5593}, peerreviewed = {Yes}, title = {{On} conditions for
asymptotic stability of dissipative infinite-dimensional systems with intermittent damping}, volume = {252}, year = {2012} } @incollection{faucris.117910584, abstract = {A decomposition-method
introduced in [Le1] is used to decompose the rigid-body motion from the elastic vibration of a slowly rotating beam. On the base of the transformed system, a controller is constructed that steers all
oscillations of the beam to rest in finite time. In addition, the beam is thereby driven to zero angular velocity. A second result is concerned with strong feed-back stabilizability.}, address =
{Berlin; Heidelberg}, author = {Leugering, Günter}, booktitle = {Optimal Control of Partial Differential Equations - Proceedings of the IFIP WG 7.2 International Conference Irsee, April 9–12, 1990},
doi = {10.1007/BFb0043223}, editor = {Prof. Karl-Heinz Hoffmann; Prof. Werner Krabs}, faupublication = {no}, isbn = {978-3-540-46883-7}, month = {Jan}, pages = {182-191}, peerreviewed = {unknown},
publisher = {New York; Springer; 1999}, series = {Lecture Notes in Control and Information Sciences}, title = {{ON} {CONTROL} {AND} {STABILIZATION} {OF} {A} {ROTATING} {BEAM} {BY} {APPLYING}
{MOMENTS} {AT} {THE} {BASE} {ONLY}}, url = {http://link.springer.com/chapter/10.1007/BFb0043223}, volume = {149}, year = {1991} } @incollection{faucris.117911464, abstract = {We consider a planar
graph representative of the reference configuration of a network of elastic prestretched strings coupled at the vertices of that graph. Some or all of the vertices may carry a point mass, and at
those nodes dry friction on the plane may occur. We briefly describe the model and some results on well-posedness and control of such systems obtained in the literature. We then introduce a dynamic
domain decomposition based on a Steklov-Poincaré-type operator. The analysis is given for the time-domain and the frequency-domain. Optimal control and problems of exact controllability are
formulated and investigated in terms of the decoupling procedure.}, address = {Basel}, author = {Leugering, Günter}, booktitle = {Control and Estimation of Distributed Parameter Systems -
International Conference in Vorau, Austria, July 14-20, 1996}, doi = {10.1007/978-3-0348-8849-3{\_}15}, editor = {W. Desch; F. Kappel; K. Kunisch}, faupublication = {no}, isbn = {978-3-0348-8849-3},
keywords = {dynamic domain decomposition;strings;networks;joint masses;dry friction;Steklov-Poincare-operators for networks;differential-delay systems;optimal control;controllability; 93C20; 93C80;
93B05; 65N55}, month = {Jan}, pages = {191-205}, peerreviewed = {unknown}, publisher = {Birkhäuser}, series = {International Series of Numerical Mathematics}, title = {{On} dynamic domain
decomposition of controlled networks of elastic strings and joint-masses}, url = {http://link.springer.com/chapter/10.1007/978-3-0348-8849-3{\_}15}, volume = {126}, year = {1998} } @article
{faucris.107401404, abstract = {This paper is concerned with the analysis of equilibrium problems for two-dimensional elastic bodies with thin rigid inclusions and cracks. Inequality-type boundary
conditions are imposed at the crack faces providing a mutual non-penetration between the crack faces. A rigid inclusion may have a delamination, thus forming a crack with non-penetration between the
opposite faces. We analyze variational and differential problem formulations. Different geometrical situations are considered, in particular, a crack may be parallel to the inclusion as well as the
crack may cross the inclusion, and also a deviation of the crack from the rigid inclusion is considered. We obtain a formula for the derivative of the energy functional with respect to the crack
length for considering this derivative as a cost functional. An optimal control problem is analyzed to control the crack growth.}, author = {Khludnev, A. M. and Leugering, Günter}, doi = {10.1002/
mma.1308}, faupublication = {yes}, journal = {Mathematical Methods in the Applied Sciences}, pages = {1955--1967}, peerreviewed = {Yes}, title = {{On} elastic bodies with thin rigid inclusions and
cracks}, volume = {33}, year = {2010} } @article{faucris.123700984, abstract = {This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of
elastic strings. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions, and, by using the methods of quasilinear hyperbolic systems, prove that for tree
networks the natural initial, boundary value problem has classical solutions existing in neighborhoods of the "stretched" equilibrium solutions. Then the local controllability of such networks near
such equilibrium configurations in a certain specified time interval is proved. Finally, it is proved that, given two different equilibrium states satisfying certain conditions, it is possible to
control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a sufficiently large time interval. © 2012
Fudan University and Springer-Verlag Berlin Heidelberg.}, author = {Leugering, Günter and Schmidt, Georg E.P.J.}, doi = {10.1007/s11401-011-0693-9}, faupublication = {yes}, journal = {Chinese Annals
of Mathematics Series B}, keywords = {Controllability; Network; Nonlinear strings; Quasilinear system of hyperbolic equations}, pages = {33-60}, peerreviewed = {Yes}, title = {{On} exact
controllability of networks of nonlinear elastic strings in 3-dimensional space}, volume = {33}, year = {2012} } @incollection{faucris.122685684, abstract = {In these notes we want to present some
control strategies for dynamic networks of strings and beams in those situations where the classical concepts of exact or approximate controllability fail. This is, in particular, the case for
networks containing circuits. Generically, a residual motion settles in such circuits, even if all nodes are subject to controls. In those situations we resort to controls which direct the flux of
energy. Using such controls in a network, we are able to steer the entire energy to preassigned parts of the structure. In practice these parts are more massive and can absorb energy more easily than
the fragile elements. We also provide numerical evidence for the control strategies discussed in these notes. The material is related to joint work with J.E. Lagnese and E.G.P.G. Schmidt, and is
essentially included [14].}, address = {Basel}, author = {Leugering, Günter}, booktitle = {Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena - International Conference in
Vorau (Austria), July 18–24, 1993}, doi = {10.1007/978-3-0348-8530-0{\_}17}, editor = {W. Desch; F. Kappel; K. Kunisch}, faupublication = {no}, isbn = {978-3-0348-8530-0}, keywords = {absorbing;
directing and nonlinear controls;disappearing solutions;numerical simulation; 39C20; 39B52; 65M06; 65M60}, month = {Jan}, pages = {301-326}, peerreviewed = {unknown}, publisher = {Birkhäuser}, series
= {ISNM International Series of Numerical Mathematics}, title = {{On} feedback controls for dynamic networks of strings and beams and their numerical simulation}, url = {http://link.springer.com/
chapter/10.1007/978-3-0348-8530-0{\_}17}, volume = {118}, year = {1994} } @inproceedings{faucris.117912344, abstract = {The authors describe a general model for vibrating networks and show how the
model applies to networks of elastic strings or Timoshenko beams. They indicate how hyperbolic energy estimates for first-order systems yield the a priori estimates needed to prove exact
controllability and stabilizability. They show how the model can be generalized to allow mixed networks of strings and beams. They also discuss generalizations of the Timoshenko beam and their
relation on the present model.}, author = {Langnese, John E. and Leugering, Günter and Schmidt, E. J. P. Georg}, booktitle = {Decision and Control}, date = {1992-12-16/1992-12-18}, doi = {10.1109/
CDC.1992.371260}, faupublication = {no}, isbn = {0-7803-0872-7}, keywords = {Yield estimation, Controllability}, month = {Jan}, pages = {3003-3008}, peerreviewed = {unknown}, publisher = {IEEE
Xplore}, title = {{ON} {HYPERBOLIC} {SYSTEMS} {ASSOCIATED} {WITH} {THE} {MODELING} {AND} {CONTROL} {OF} {VIBRATING} {NETWORKS}}, url = {http://ieeexplore.ieee.org/document/371260/}, year = {1992} }
@inproceedings{faucris.120849784, abstract = {We consider a system of 1-d-diffusion-convection equations on graphs as being representative of flow or transport problems in channel, pipeline or
root-systems appearing in many applications. Such systems are subject to controls the effect of which one wants to optimize. In these notes we discuss the mathematical framework of typical optimal
control problems of such systems on graphs. We focus on singular perturbations with respect to diffusion and on the numerical implementation.}, author = {Leugering, Günter and Fischer, Thorsten},
faupublication = {yes}, month = {Jan}, pages = {69-97}, peerreviewed = {unknown}, title = {{On} instantaneous control of singularly perturbed hyperbolic equations on graphs}, volume = {219}, year =
{2001} } @incollection{faucris.106894304, address = {Hildesheim}, author = {Rathmann, Wigand and von Schroeders, Nicolai}, booktitle = {Medien im Mathematikunterricht}, editor = {Stephanie Gleich},
faupublication = {yes}, isbn = {978-3-88120-842-0}, pages = {39-56}, peerreviewed = {unknown}, publisher = {Franzbecker}, series = {MaMut Materialien für den Mathematikunterricht}, title = {{Online}-
{Auswertungsverfahren} {STACK}}, volume = {6}, year = {2018} } @inproceedings{faucris.111074964, abstract = {We consider a network of Euler-Bernoulli- and Rayleigh-beams. For the sake of simplicity,
we concentrate on scalar displacements coupled to torsion. We show that the model is well-posed in appropriate ramification spaces. We then describe a dynamic nonoverlapping domain decomposition
procedure of the network into its individual edges and provide a proof of convergence. Further, we formulate typical optimal control problems, related to exact controllability. The optimality system
is solved using conjugate gradients. Various numerical examples illustrate the method.}, author = {Leugering, Günter and Rathmann, Wigand}, faupublication = {yes}, keywords = {Euler-Bernoulli- and
Rayleigh beams;network models;nonoverlapping domain decompositions;optimal controls}, month = {Jan}, pages = {199-232}, peerreviewed = {unknown}, title = {{On} modeling, analysis and simulation of
optimal control problems for dynamic networks of {Euler}-{Bernoulli}- and {Rayleigh}-beams}, volume = {218}, year = {2001} } @incollection{faucris.114041004, address = {New York}, author = {Rathmann,
Wigand and Leugering, Günter}, booktitle = {Control of Nonlinear Distributed Parameter Systems}, editor = {Chen G, Lasiecka I, Zhou J}, faupublication = {no}, peerreviewed = {unknown}, publisher =
{Marcel Dekker Inc}, series = {Lecture Notes in Pure and Applied Mathematics Series}, title = {{On} {Modelling}, {Analysis} and {Simulation} of {Optimal} {Control} {Problems} of {Euler}-{Bernoulli}-
and {Rayleigh}-{Beams}}, year = {2001} } @article{faucris.118521744, abstract = {A shape optimization problem in three spatial dimensions for an elasto-dynamic piezoelectric body coupled to an
acoustic chamber is introduced. Well-posedness of the problem is established and first order necessary optimality conditions are derived in the framework of the boundary variation technique. In
particular, the existence of the shape gradient for an integral shape functional is obtained, as well as its regularity, sufficient for applications e.g. in modern loudspeaker technologies. The shape
gradients are given by functions supported on the moving boundaries. The paper extends results obtained by the authors in (Math. Methods Appl. Sci. 33(17):2118–2131, 2010) where a similar problem was
treated without acoustic coupling.}, author = {Leugering, Günter and Novotny, A. A. and Menzala, G. Perla and Sokolowski, J.}, doi = {10.1007/s00245-011-9148-7}, faupublication = {yes}, journal =
{Applied Mathematics and Optimization}, pages = {441--466}, peerreviewed = {Yes}, title = {{On} {Shape} {Optimization} for an {Evolution} {Coupled} {System}}, volume = {64}, year = {2011} } @article
{faucris.117913444, abstract = {We study the limiting behavior of an optimal control problem for a linear elliptic equation subject to control and state constraints. Each constituent of the
mathematical description of such an optimal control problem may depend on a small parameter epsilon. We study the limit of this problem when epsilon --> 0 in the framework of variational
S-convergence which generalizes the concept of F-convergence. We also introduce the notion of G*-convergence generalizing the concept of G-convergence to operators. with constraints. We show
convergence of the sequence of optimal control problems and identify its limit. We then apply the theory to an elliptic problem on a perforated domain.}, author = {Leugering, Günter and Kogut, Peter
I.}, doi = {10.4171/ZAA/1023}, faupublication = {yes}, journal = {Zeitschrift für Analysis und ihre Anwendungen}, keywords = {homogenization;S-convergence;optimal control}, month = {Jan}, pages =
{395-429}, peerreviewed = {Yes}, title = {{On} {S}-homogenization of an optimal control problem with control and state constraints}, url = {https://www.ems-ph.org/journals/show{\_}abstract.php?issn=
0232-2064&vol=20&iss=2&rank=9}, volume = {20}, year = {2001} } @incollection{faucris.106691024, address = {Berlin}, author = {Hante, Falk and Amin, Saurabh and Bayen, Alexandre M.}, booktitle =
{Hybrid systems: computation and control}, doi = {10.1007/978-3-540-78929-1{\_}44}, faupublication = {yes}, pages = {602--605}, peerreviewed = {unknown}, publisher = {Springer}, series = {Lecture
Notes in Comput. Sci.}, title = {{On} stability of switched linear hyperbolic conservation laws with reflecting boundaries}, volume = {4981}, year = {2008} } @article{faucris.115625004, abstract =
{Using the concept of contingent epiderivative, we generalize the notion of subdifferential to a cone-convex set-valued map. Properties of the subdifferential are presented and an optimality
condition is discussed.}, author = {Jahn, Johannes and Baier, J.}, faupublication = {yes}, journal = {Journal of Optimization Theory and Applications}, keywords = {Convex analysis; Set-valued
analysis; Subdifferentials; Vector optimization}, note = {UnivIS-Import:2015-03-05:Pub.1999.nat.dma.pama21.onsubd}, pages = {233-240}, peerreviewed = {Yes}, title = {{On} {Subdifferentials} of {Set}-
{Valued} {Maps}}, volume = {100}, year = {1999} } @article{faucris.117913664, abstract = {In this paper a general linear model for vibrating networks of one-dimensional elements is derived. This is
applied to various situations including nonplanar networks of beams modelled by a three-dimensional variant on the Timoshenko beam, described for the first time in this paper. The existence and
regularity of solutions is established for all the networks under consideration. The methods of first-order hyperbolic systems are used to obtain estimates from which exact controllability follows
for networks containing no closed loops.}, author = {Langnese, John E. and Leugering, Günter and Schmidt, E. J. P. Georg}, doi = {10.1017/S0308210500029206}, faupublication = {no}, journal =
{Proceedings of the Royal Society of Edinburgh Section A-Mathematics}, month = {Jan}, pages = {77-104}, peerreviewed = {Yes}, title = {{ON} {THE} {ANALYSIS} {AND} {CONTROL} {OF} {HYPERBOLIC}
{SYSTEMS} {ASSOCIATED} {WITH} {VIBRATING} {NETWORKS}}, url = {https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/
on-the-analysis-and-control-of-hyperbolic-systems-associated-with-vibrating-networks/92C650CDE4C03DF26D6009C6E5B61A6E}, volume = {124}, year = {1994} } @incollection{faucris.123354044, address =
{Berlin}, author = {Jahn, Johannes and Klose, Jürgen and Merkel, A.}, booktitle = {Advances in Optimization - Proceedings, Lambrecht, FRG, 1991}, editor = {W. Oettli, D. Pallaschke}, faupublication =
{yes}, note = {UnivIS-Import:2015-04-20:Pub.1992.nat.dma.pama21.onthea}, pages = {478-491}, peerreviewed = {unknown}, publisher = {Springer}, series = {Lecture Notes in Economics and Mathematical
Systems}, title = {{On} the {Application} of a {Method} of {Reference} {Point} {Approximation} to {Bicriterial} {Optimization} {Problems} in {Chemical} {Engineering}}, volume = {382}, year = {1992} }
@article{faucris.122686784, abstract = {A problem of image registration is considered in the context of optimal mass transportation. The properties and limitations of an optimal image transportation
are analyzed. A modified formulation of this approach is proposed in order to overcome the morphing effect. Finally, a fast and simple scale-space approach for the new formulation is introduced, and
numerical examples are presented.}, author = {Museyko, Oleg and Stiglmayr, Michael and Klamroth, Kathrin and Leugering, Günter}, doi = {10.1137/080721522}, faupublication = {yes}, journal = {Siam
Journal on Imaging Sciences}, keywords = {image registration;image morphing;optimal mass transportation;Monge-Kantorovich problem; 94A08; 92C55; 49N99}, month = {Jan}, pages = {1068-1097},
peerreviewed = {Yes}, title = {{On} the {Application} of the {Monge}-{Kantorovich} {Problem} to {Image} {Registration}}, url = {http://epubs.siam.org/doi/abs/10.1137/080721522?journalCode=sjisbi},
volume = {2}, year = {2009} } @article{faucris.120147984, abstract = {This paper works out connections between semidefinite optimization and vector optimization. It is shown that well-known
semidefinite optimization problems are scalarized versions of a general vector optimization problem. This scalarization leads to the minimization of the trace or the maximal eigenvalue. © Taru
Publications.}, author = {Jahn, Johannes and Carosi, L. and Martein, L.}, doi = {10.1080/09720502.2003.10700341}, faupublication = {yes}, journal = {Journal of Interdisciplinary Mathematics}, note =
{UnivIS-Import:2015-03-09:Pub.2003.nat.dma.pama21.xxx}, pages = {219-229}, peerreviewed = {unknown}, title = {{On} the {Connections} between {Semidefinite} {Optimization} and {Vector}
{Optimization}}, volume = {6}, year = {2003} } @inproceedings{faucris.117915204, abstract = {The authors introduce nonlinear equations describing the vibrations in space of networks of elastic
strings and beams. Linearization about an equilibrium configuration yields a hyperbolic system of linear equations coupled by conditions at the multiple nodes where several network members meet. The
authors study the controllability of these systems by controls exercised at some nodes. The multiplier method yields a priori inequalities guaranteeing exact controllability, and hence
stabilizability, for certain rudimentary networks.}, author = {Leugering, Günter and Schmidt, E. J. P. Georg}, booktitle = {Decision and Control, 1989., Proceedings of the 28th IEEE Conference}, date
= {1989-12-13/1989-12-15}, doi = {10.1109/CDC.1989.70580}, editor = {IEEE}, faupublication = {no}, keywords = {Nonlinear equations; Potential energy; Mathematics; Controllability; Control systems;
Mathematical model; Kinetic energy; Partial differential equations; Statistics; Councils}, month = {Jan}, pages = {2287-2290}, peerreviewed = {unknown}, publisher = {IEEE Xplore}, title = {{ON} {THE}
{CONTROL} {OF} {NETWORKS} {OF} {VIBRATING} {STRINGS} {AND} {BEAMS}}, url = {http://ieeexplore.ieee.org/document/70580/}, year = {1989} } @article{faucris.119866164, author = {Jahn, Johannes},
faupublication = {yes}, journal = {Journal of Multi-Criteria Decision Analysis}, note = {UnivIS-Import:2015-03-05:Pub.1997.nat.dma.pama21.onthed}, pages = {17-24}, peerreviewed = {Yes}, title = {{On}
the {Determination} of {Minimal} {Facets} and {Edges} of a {Polyhedral} {Set}}, volume = {6}, year = {1997} } @article{faucris.118531644, abstract = {We investigate the occurrence of
self-penalization in topology optimization problems for piezoceramicmechanical composites. Our main goal is to give physical interpretations for this phenomenon, i.e., to study the question why for
various problems intermediate material values are not optimal in the absence of explicit penalization of the pseudo densities. In order to investigate this effect numerical experiments for several
static and/or dynamic actuator and sensor objective functions are performed and their respective results are compared. The objective functions are mean transduction, displacement, sound power,
electric potential, electric energy, energy conversion and electric power.}, author = {Wein, Fabian and Kaltenbacher, Manfred and Kaltenbacher, Barbara and Leugering, Günter and Bänsch, Eberhard and
Schury, Fabian}, doi = {10.1007/s00158-010-0570-2}, faupublication = {yes}, journal = {Structural and Multidisciplinary Optimization}, pages = {405-417}, peerreviewed = {Yes}, title = {{On} the
effect of self-penalization of piezoelectric composites in topology optimization}, volume = {43}, year = {2011} } @article{faucris.245142909, abstract = {This article addresses the numerical
simulation and optimization of the optical properties of mono-layered nano-particulate films. The particular optical property of interest is the so-called haze factor, for which a model close to an
experimental setup is derived. The numerical solution becomes very involved due to the resulting size of computational domain in comparison to the wave length, rendering the direct simulation method
infeasible. As a remedy, a hybrid method is suggested, which in essence consistently combines analytical solutions for the far field with finite element-based solutions for the near field. Using a
series of algebraic reformulations, a model with an off- and online component is developed, which results in the computational complexity being reduced by several orders of magnitude. Based on the
suggested hybrid numerical scheme, which is not limited to the haze factor as objective function, structural optimization problems covering geometrical and topological parametrizations are
formulated. These allow the influence of different particle arrangements to be studied. The article is complemented by several numerical experiments underlining the strength of the method.}, author =
{Semmler, Johannes and Stingl, Michael}, doi = {10.1007/s00158-020-02754-6}, faupublication = {yes}, journal = {Structural and Multidisciplinary Optimization}, keywords = {Finite element method;
Gradient-based optimization; Haze factor; Maxwell’s equation; Particle scattering; Vector spherical harmonics}, note = {CRIS-Team Scopus Importer:2020-11-13}, peerreviewed = {Yes}, title = {{On} the
efficient optimization of optical properties of particulate monolayers by a hybrid finite element approach}, year = {2020} } @article{faucris.119552224, abstract = {We present an asymptotic approach
to find optimal rotations of orthotropic material inclusions inside an isotropic linear elastic matrix. We compute approximate optimal solutions with respect to compliance and a stress based cost
functional. We validate the local and global quality of the candidate solutions by means of finite element based parametric optimization algorithms. In particular, we devise a lower bound algorithm
based on the free material optimization approach. Several numerical experiments are performed for different traction scenario}, author = {Schury, Fabian and Greifenstein, Jannis and Leugering, Günter
and Stingl, Michael}, doi = {10.1007/s11081-014-9262-x}, faupublication = {yes}, journal = {Optimization and Engineering}, pages = {225-246}, peerreviewed = {Yes}, title = {{On} the efficient
solution of a patch problem with multiple elliptic inclusions}, volume = {16}, year = {2015} } @article{faucris.121727144, abstract = {The practice of production of composite materials with thin
fibers requires a correct description of problems of the equilibrium of elastic bodies containing thin rigid inclusions. Of large importance is also an adequate description of the interaction of thin
inclusions with cracks in an elastic body. As is known, inclusions can delaminate from the matrix, thereby forming a crack in the elastic body. A traditional approach to the description of cracks in
elastic bodies consists in the use of equality-type boundary conditions at the crack faces. This approach frequently leads to unphysical results; for example, the corresponding solutions can yield a
mutual penetration of the crack faces (see, e.g., [1, 2]). In the recent years, there was performed a wide cycle of investigations in the theory of cracks with boundary conditions corresponding to
the mutual nonpenetration of the crack edges. In this case, the boundary conditions at the crack faces have the form of a set of equalities and inequalities. For acquaintance with such models, one
can turn to [3, 4], where numerous appropriate references can be found. In this works, we suggest a model of an elastic body containing thin rigid inclusions with a property of delamination. For
describing the delamination, we use inequality-type boundary conditions eliminating the mutual penetration of the crack faces.}, author = {Leugering, Günter and Khludnev, A. M.}, doi = {10.1134/
S1028335810010040}, faupublication = {yes}, journal = {Doklady Physics}, pages = {18--22}, peerreviewed = {Yes}, title = {{On} the equilibrium of elastic bodies containing thin rigid inclusions},
volume = {55}, year = {2010} } @misc{faucris.107879244, author = {Jahn, Johannes and Eibenstein, F. and Leitmann, G.}, faupublication = {no}, note =
{UnivIS-Import:2015-03-05:Pub.1984.nat.dma.pama21.ontheg}, peerreviewed = {automatic}, title = {{On} the {Global} {Asymptotic} {Stabilization} of {Nonlinear} {Systems}}, year = {1984} } @article
{faucris.124195324, abstract = {In this article the authors continue the discussion in [9] about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary
measurements. In the present paper we prove the identifiability of varying densities of a planar tree-like network of strings along with the complete information on the graph, i.e. the lengths of the
edges, the edge degrees and the angles between neighbouring edges. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincare operator for the
dynamic wave equation on the tree. The general result is obtained by a peeling argument which reduces the inverse problem layer-by-layer from the leaves to the clamped root of the tree. (C) 2015
WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim}, author = {Avdonin, Sergei and Abdon, Choque Rivero and Leugering, Günter and Mikhaylov, Victor}, doi = {10.1002/zamm.201400126}, faupublication = {yes},
journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik}, keywords = {Inverse problem;planar tree of strings;Titchmarch-Weyl-matrix;dynamic Steklov-Poincare operator;leaf-peeling
method}, pages = {1490-1500}, peerreviewed = {Yes}, title = {{On} the inverse problem of the two-velocity tree-like graph}, volume = {95}, year = {2015} } @article{faucris.223557645, abstract = {An
example by Bastin and Coron illustrates that the boundary stabilization of 1-d hyperbolic systems with certain source terms is only possible if the length of the space interval is sufficiently small.
We show that related phenomena also occur for networks of vibrating strings that are governed by the wave equation with a certain source term. It turns out that for a tree of strings with Neumann
velocity feedback control at one boundary node and a homogeneous Dirichlet boundary condition at at least one boundary node and homogeneous Dirichlet or Neumann conditions at the other boundary
nodes, boundary feedback stabilization is not possible if one of the strings is sufficiently long. However, if the number of strings in the tree is sufficiently large, also for arbitrarily short
strings for certain parameters in the source term stabilization is not possible. The wave equation with source term that we consider is equivalent to a certain 2 ×2 system. For the examples that
illustrate the limits of stabilizability, the matrix of the source term is not positive definite. However if the system parameters are chosen in such a way that the matrix is positive semi-definite,
the tree of strings can be stabilized exponentially fast by the boundary feedback control for arbitrary long space intervals.}, author = {Gugat, Martin and Gerster, Stephan}, doi = {10.1016/
j.sysconle.2019.104494}, faupublication = {yes}, journal = {Systems & Control Letters}, keywords = {Limits of stabilizability; Neumann feedback; Source terms; Stabilization; String networks}, note =
{CRIS-Team Scopus Importer:2019-08-02}, peerreviewed = {Yes}, title = {{On} the limits of stabilizability for networks of strings}, volume = {131}, year = {2019} } @article{faucris.120144684,
abstract = {In this paper we investigate how to determine optimal locations of the microwave antennas being circularly ordered in a hyperthermia device. The heated area containing the tumor should
have minimal volume. Based on a simple geometric model for the two and three dimensional case we develop algorithms for the computation of these volumes and present numerical results for the optimal
location}, author = {Hutterer, Angelika and Jahn, Johannes}, faupublication = {yes}, journal = {Or Spectrum}, keywords = {Antenna optimization; Cancer medicine; Nonlinear programming}, note =
{UnivIS-Import:2015-03-09:Pub.2003.nat.dma.pama21.onthel}, pages = {397-412}, peerreviewed = {Yes}, title = {{On} the location of antennas for treatment planning in hyperthermia}, volume = {25}, year
= {2003} } @article{faucris.108612284, abstract = {Congrès
International conference on control of distributed parameter systems. 4 (1988)
1989, vol. 91, pp. 249-261 (12 ref.)}, author = {Leugering, Günter}, faupublication = {no}, journal = {International Series of Numerical Mathematics}, month = {Jan}, pages = {249-261}, peerreviewed =
{unknown}, title = {{ON} {THE} {REACHABILITY} {PROBLEM} {OF} {A} {VISCOELASTIC} {BEAM} {DURING} {A} {SLEWING} {MANEUVER}}, volume = {91}, year = {1989} } @article{faucris.117458704, abstract = {We
consider the isothermal Euler equations without friction that simulate gas flow through a pipe. We consider the problem of boundary stabilisation of this system locally around a given stationary
state. We present a feedback law that is linear in the physical variables and yields exponential decay of the system state. For the numerical solution of hyperbolic systems of conservation laws, the
Jin-Xin relaxation scheme can be used. Therefore, we also consider the boundary stabilisation of the relaxation system by the linear Riemann feedback and present numerical examples that show the
rapid exponential decay of the stabilised system. © 2012 Taylor & Francis Group, LLC.}, author = {Hirsch-Dick, Markus and Gugat, Martin and Herty, Michael and Steffensen, Sonja}, doi = {10.1080/
00207179.2012.703787}, faupublication = {yes}, journal = {International Journal of Control}, keywords = {boundary feedback stabilisation; conservation laws; isothermal Euler equations; Lyapunov
function; relaxation scheme}, note = {UnivIS-Import:2015-03-09:Pub.2012.nat.dma.zentr.onther}, pages = {1766-1778}, peerreviewed = {Yes}, title = {{On} the relaxation approximation of boundary
control of the isothermal {Euler} equations}, volume = {85}, year = {2012} } @article{faucris.111754104, author = {Hante, Falk}, doi = {10.1002/pamm.201610380}, faupublication = {yes}, journal =
{Proceedings in Applied Mathematics and Mechanics}, pages = {783--784}, peerreviewed = {Yes}, title = {{On} the relaxation gap for {PDE} mixed-integer optimal control problems}, volume = {16}, year =
{2016} } @article{faucris.286938922, abstract = {We consider Helmholtz problems in two and three dimensions. The topological sensitivity of a given cost function J(uϵ) with respect to a small hole Bϵ
around a given point x0 ϵ Bϵ ⊂ ω depends on various parameters, like the frequency k chosen or certain material parameters or even the shape parameters of the hole Bϵ. These parameters are either
deliberately chosen in a certain range, as, e.g., the frequencies, or are known only up to some bounds. The problem arises as to whether one can obtain a uniform design using the topological
gradient. We show that for 2-d and 3-d Helmholtz problems such a robust design is achievable.}, author = {Leugering, Günter and Novotny, Antonio André and Sokolowski, Jan}, doi = {10.2478/
candc-2022-0015}, faupublication = {yes}, journal = {Control and Cybernetics}, keywords = {complex variables; Helmholtz problem; inverse problems; numerical methods; shape optimization; topological
derivative}, note = {CRIS-Team Scopus Importer:2022-12-23}, pages = {227-248}, peerreviewed = {Yes}, title = {{On} the robustness of the topological derivative for {Helmholtz} problems and
applications}, volume = {51}, year = {2022} } @article{faucris.122687444, abstract = {We consider a star-graph as an examplary network, with elastic strings stretched along the edges. The network is
allowed to perform out-of-the plane displacements. We consider such networks as being controlled at its simple nodes via Dirichlet conditions. The objective is to steer given initial data to final
target data in a given time T with minimal control costs. This problem is discussed in the continuous as well as in the discrete case. We discuss an iterative domain decomposition technique and its
discrete analogue. We prove convergence and show some numerical results. (C) 2000 Elsevier Science B.V. All rights reserved.}, author = {Leugering, Günter}, doi = {10.1016/S0377-0427(00)00307-1},
faupublication = {no}, journal = {Journal of Computational and Applied Mathematics}, keywords = {domain decomposition;networks of strings;optimal control;semi-discrete models}, pages = {133-157},
peerreviewed = {Yes}, title = {{On} the semi-discretization of optimal control problems for networks of elastic strings: global optimality systems and domain decomposition}, url = {http://
www.sciencedirect.com/science/article/pii/S0377042700003071}, volume = {120}, year = {2000} } @incollection{faucris.118857684, address = {Heidelberg}, author = {Jahn, Johannes}, booktitle =
{Operations Research '93}, editor = {A. Bachem, U. Derigs, M. Jünger, R. Schrader}, faupublication = {yes}, note = {UnivIS-Import:2015-04-20:Pub.1994.nat.dma.pama21.onthes}, pages = {280-283},
peerreviewed = {unknown}, publisher = {Physica}, title = {{On} the {Solution} of {Linear} {Vector} {Optimization} {Problems} in the {Image} {Space}}, year = {1994} } @article{faucris.213483552,
abstract = {We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over
the finite time interval (0, T) that depends on the boundary traces of the solution. If the time horizon T is sufficiently large, the solution of the dynamic optimal boundary control problem can be
approximated by the solution of a steady state optimization problem. We show that for T -> infinity the approximation error converges to zero in the sense of the norm in L-2 (0, 1) with the rate 1/T,
if the time interval (0, T) is transformed to the fixed interval (0, 1). Moreover, we show that also for optimal boundary control problems with integer constraints for the controls the turnpike
phenomenon occurs. In this case the steady state optimization problem also has the integer constraints. If T is sufficiently large, the integer part of each solution of the dynamic optimal boundary
control problem with integer constraints is equal to the integer part of a solution of the static problem. A numerical verification is given for a control problem in gas pipeline operations.}, author
= {Gugat, Martin and Hante, Falk}, doi = {10.1137/17M1134470}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, month = {Jan}, note = {CRIS-Team WoS Importer:2019-03-15},
pages = {264-289}, peerreviewed = {Yes}, title = {{ON} {THE} {TURNPIKE} {PHENOMENON} {FOR} {OPTIMAL} {BOUNDARY} {CONTROL} {PROBLEMS} {WITH} {HYPERBOLIC} {SYSTEMS}}, volume = {57}, year = {2019} }
@article{faucris.122593064, abstract = {The paper concerns the analysis of equilibrium problems for 2D elastic bodies with thin inclusions modeled in the framework of Timoshenko beams. The first
focus is on the well-posedness of the model problem in a variational setting. Then delaminations of the inclusions are considered, forming a crack between the elastic body and the inclusion.
Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. The corresponding variational formulations together with weak and strong solutions
are discussed. The model contains various physical parameters characterizing the mechanical properties of the inclusion, such as flexural and shear stiffness. The paper provides an asymptotic
analysis of such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non-penetration conditions, respectively.
Finally, exemplary networks of Timoshenko beams are considered as inclusions as well.}, author = {Khludnev, A. M. and Leugering, Günter}, doi = {10.1177/1081286513505106}, faupublication = {yes},
journal = {Mathematics and Mechanics of Solids}, keywords = {Thin elastic inclusion;Timoshenko beams;crack;delamination;non-penetration boundary condition}, pages = {495-511}, peerreviewed = {Yes},
title = {{On} {Timoshenko} thin elastic inclusions inside elastic bodies}, volume = {20}, year = {2015} } @article{faucris.107412624, abstract = {In this paper, we study an optimal control problem
for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for
the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body Ω is taken as a boundary control. Because the initial
boundary value problem of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational
solutions. Using the convergence concept in variable spaces and following the direct method in calculus of variations, we prove the existence of optimal and approximate solutions to the optimal
control problem under rather general assumptions on the quasistatic evolution of damage.}, author = {Kogut, Peter I. and Leugering, Günter}, doi = {10.1002/mma.3257}, faupublication = {yes}, journal
= {Mathematical Methods in the Applied Sciences}, pages = {n/a}, peerreviewed = {Yes}, title = {{Optimal} and approximate boundary controls of an elastic body with quasistatic evolution of damage},
year = {2014} } @book{faucris.122947704, abstract = {This brief considers recent results on optimal control and stabilization of systems governed by hyperbolic partial differential equations,
specifically those in which the control action takes place at the boundary. The wave equation is used as a typical example of a linear system, through which the author explores initial boundary value
problems, concepts of exact controllability, optimal exact control, and boundary stabilization. Nonlinear systems are also covered, with the Korteweg-de Vries and Burgers Equations serving as
standard examples. To keep the presentation as accessible as possible, the author uses the case of a system with a state that is defined on a finite space interval, so that there are only two
boundary points where the system can be controlled. Graduate and post-graduate students as well as researchers in the field will find this to be an accessible introduction to problems of optimal
control and stabilization.}, address = {Basel}, author = {Gugat, Martin}, doi = {10.1007/978-3-319-18890-4}, edition = {1}, faupublication = {yes}, isbn = {978-3-319-18889-8}, keywords = {Systems
Theory, Control; Partial Differential Equations; Calculus of Variations and Optimal Control; Optimization}, peerreviewed = {unknown}, publisher = {Birkhäuser}, series = {SpringerBriefs in Control,
Automation and Robotics}, title = {{Optimal} {Boundary} {Control} and {Boundary} {Stabilization} of {Hyperbolic} {Systems}}, url = {http://www.springer.com/us/book/9783319188898}, year = {2015} }
@incollection{faucris.119025984, abstract = {
In active flood hazard mitigation, lateral flow withdrawal is used to reduce the impact of flood waves in rivers. Through emergency side channels, lateral outflow is generated. The optimal outflow
controls the flood in such a way that the cost of the created damage is minimized. The flow is governed by a networked system of nonlinear hyperbolic partial differential equations, coupled by
algebraic node conditions. Two types of integrals appear in the objective function of the corresponding optimization problem: Boundary integrals (for example, to measure the amount of water that
flows out of the system into the floodplain) and distributed integrals.
For the evaluation of the derivative of the objective function, we introduce an adjoint backwards system. For the numerical solution we consider a discretized system with a consistent discretization
of the continuous adjoint system, in the sense that the discrete adjoint system yields the derivatives of the discretized objective function. Numerical examples are included.
}, address = {Basel}, author = {Gugat, Martin}, booktitle = {International Series of Numerical Mathematics}, doi = {10.1007/978-3-7643-7721-2{\_}4}, editor = {Karl-Heinz Hoffmann, Günter Leugering,
Michael Hintermüller}, faupublication = {yes}, isbn = {978-3-7643-7721-2}, keywords = {St. Venant equations; subcritical states; adjoint system; optimal boundary control; necessary optimality
conditions; classical solutions}, note = {UnivIS-Import:2015-04-20:Pub.2007.nat.dma.zentr.optima}, pages = {69-94}, peerreviewed = {unknown}, publisher = {Birkhäuser}, series = {Control of Coupled
Partial Differential Equations}, title = {{Optimal} {Boundary} {Control} in {Flood} {Management}}, url = {http://link.springer.com/chapter/10.1007/978-3-7643-7721-2{\_}4}, volume = {155}, year =
{2007} } @article{faucris.121019404, abstract = {The problem to control a finite string to the zero state in finite time from a given initial state by controlling the state at the two boundary points
in such a way that the controls generate a continuous state is considered. The corresponding optimal control problem where the objective function is the L ^2 norm of the derivatives of the controls
is solved explicitly in the sense that controls that are successful and minimize at the same time the objective function are determined as functions of the initial state. © 2006 WILEY-VCH Verlag GmbH
& Co. KGaA.}, author = {Gugat, Martin}, doi = {10.1002/zamm.200410236}, faupublication = {yes}, journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik}, keywords = {Analytic solution;
Continuous state; Exact controllability; Optimal boundary control; Wave equation}, note = {UnivIS-Import:2015-03-09:Pub.2006.nat.dma.lama1.optima}, pages = {134-150}, peerreviewed = {Yes}, title =
{{Optimal} boundary control of a string to rest in finite time with continuous state}, url = {http://www3.interscience.wiley.com/search/allsearch?WISsearch2=Gugat&WISindexid2=WISall&mode=quicksearch&
contentTitle=ZAMM&WISindexid1=issn&WISsearch1=1521-4001&articleGo.x=0&articleGo.y=0}, volume = {86}, year = {2006} } @inproceedings{faucris.118211764, abstract = {We investigate a new approach for
solving boundary control problems for dynamical systems that are governed by transport equations, when the control function is restricted to binary values. We consider these problems as hybrid
dynamical systems embedded with partial differential equations and present an optimality condition based on sensitivity analysis for the objective when the dynamics are governed by semilinear
convection-reaction equations. These results make the hybrid problem accessible for continuous non-linear optimization techniques. For the computation of optimal solution approximations, we propose
using meshfree solvers to overcome essential difficulties with numerical dissipation for these distributed hybrid systems. We compare results obtained by the proposed method with solutions taken from
a mixed inter programming formulation of the control problem.}, address = {Berlin Heidelberg}, author = {Hante, Falk and Leugering, Günter}, booktitle = {Lecture Notes in Computer Science}, date =
{2009-04-13/2009-04-15}, doi = {10.1007/978-3-642-00602-9{\_}15}, editor = {Rupak Majumdar; Paulo Tabuada}, faupublication = {yes}, isbn = {978-3-642-00602-9}, month = {Jan}, note =
{UnivIS-Import:2015-04-16:Pub.2009.nat.dma.lama1.optima}, pages = {209-222}, peerreviewed = {Yes}, publisher = {Springer-verlag}, title = {{Optimal} {Boundary} {Control} of {Convection}-{Reaction}
{Transport} {Systems} with {Binary} {Control} {Functions}}, url = {http://link.springer.com/chapter/10.1007/978-3-642-00602-9{\_}15}, venue = {San Francisco, CA, USA}, volume = {5469}, year = {2009}
} @article{faucris.111572604, abstract = {In optimal control problems frequently pointwise control constraints appear. We consider a finite string that is fixed at one end and controlled via
Dirichlet conditions at the other end with a given upper bound M for the L ^∞-norm of the control. The problem is to control the string to the zero state in a given finite time. If M is too small, no
feasible control exists. If M is large enough, the optimal control problem to find an admissible control with minimal L ^2-norm has a solution that we present in this paper. A finite difference
discretization of the optimal control problem is considered and we prove that for Lipschitz continuous data the discretization error is of the order of the stepsize. © 2009 Springer Science+Business
Media, LLC.}, author = {Gugat, Martin and Grimm, Volker}, doi = {10.1007/s10589-009-9289-7}, faupublication = {yes}, journal = {Computational Optimization and Applications}, keywords = {Boundary
control; Control constraints; Dirichlet control; Discretization; Discretization error; Finite differences; Optimal control; Wave equation}, note =
{UnivIS-Import:2015-03-09:Pub.2011.nat.dma.zentr.optima}, pages = {123-147}, peerreviewed = {Yes}, title = {{Optimal} boundary control of the wave equation with pointwise control constraints}, url =
{http://springerlink.com/content/k6345j348j147564/}, volume = {49}, year = {2011} } @article{faucris.111889844, abstract = {We consider a finite string that is fixed at one end and subject to a
feedback control at the other end which is allowed to move. We show that the behaviour is similar to the situation where both ends are fixed: As long as the movement is not too fast, the energy
decays exponentially and for a certain parameter in the feedback law it vanishes in finite time. We consider movements of the boundary that are continuously differentiable with a derivative whose
absolute value is smaller than the wave speed. We solve a problem of worst-case optimal feedback control, where the parameter in the feedback law is chosen such that the worst-case L p -norm of the
space derivative at the fixed end of the string is minimized (p ∈ 1, ∞)). We consider the worst case both with respect to the initial conditions and with respect to the boundary movement. It turns
out that the parameter for which the energy vanishes in finite time is optimal in this sense for all p. © The author 2007. Published by Oxford University Press on behalf of the Institute of
Mathematics and its Applications. All rights reserved.}, author = {Gugat, Martin}, doi = {10.1093/imamci/dnm014}, faupublication = {yes}, journal = {IMA Journal of Mathematical Control and
Information}, keywords = {Feedback; Moving boundary; Optimal boundary control; Optimal control of PDEs; PDE constrained optimization; Wave equation}, note =
{UnivIS-Import:2015-03-09:Pub.2008.nat.dma.zentr.optima{\_}5}, pages = {111-121}, peerreviewed = {Yes}, title = {{Optimal} boundary feedback stabilization of a string with moving boundary}, url =
{http://imamci.oxfordjournals.org/cgi/content/abstract/dnm014?ijkey=sCu0noGNy5dtzdd&keytype=ref}, volume = {25}, year = {2008} } @article{faucris.111518264, abstract = {We consider traffic flow
models for road networks where the flow is controlled at the nodes of the network. For the analytical and numerical optimization of the control, the knowledge of the gradient of the objective
functional is useful. The adjoint calculus introduced below determines the gradient in two ways. We derive the adjoint equations for the continuous traffic flow network model and derive also the
adjoint equations for a discretized model. Numerical examples for the solution of problems of optimal control for traffic flow networks are presented. © 2005 Springer Science+Business Media, Inc.},
author = {Gugat, Martin and Herty, Michael and Klar, Axel and Leugering, Günter}, doi = {10.1007/s10957-005-5499-z}, faupublication = {yes}, journal = {Journal of Optimization Theory and
Applications}, keywords = {Adjoint systems; Hyperbolic systems; Networks; Traffic flows}, note = {UnivIS-Import:2015-03-09:Pub.2005.nat.dma.lama1.optima}, pages = {589-616}, peerreviewed = {Yes},
title = {{Optimal} {Control} for {Traffic} {Flow} {Networks}}, url = {http://springerlink.metapress.com/app/home/contribution.asp?wasp=a48c9ab2fb6849c59cda936123adab85&referrer=parent&backto=
issue,6,12;journal,1,105;linkingpublicationresults,1:104801,1}, volume = {126}, year = {2005} } @article{faucris.124195544, abstract = {In this article we study an optimal control problem for a
nonlinear monotone Dirichlet problem where the controls are taken as matrix-valued coefficients in L-infinity (Omega; R-NxN). For the exemplary case of a tracking cost functional, we derive first
order optimality conditions. This first part is concerned with the general case of matrix-valued coefficients under some hypothesis, while the second part focuses on the special class of diagonal
matrices.}, author = {Kogut, Peter I. and Kupenko, Ol'Ga P. and Leugering, Günter}, doi = {10.4171/ZAA/1530}, faupublication = {yes}, journal = {Zeitschrift für Analysis und ihre Anwendungen},
keywords = {Nonlinear monotone Dirichlet problem;control in coefficients;adjoint equation}, month = {Jan}, pages = {85-108}, peerreviewed = {Yes}, title = {{Optimal} {Control} in {Matrix}-{Valued}
{Coefficients} for {Nonlinear} {Monotone} {Problems}: {Optimality} {Conditions} {I}}, volume = {34}, year = {2015} } @article{faucris.107271824, abstract = {In this paper we study an optimal control
problem for a nonlinear monotone Dirichlet problem where the controls are taken as the matrix-valued coefficients in L-infinity(Omega; R-NxN). Given a suitable cost function, the objective is to
provide a substantiation of the first order optimality conditions using the concept of convergence in variable spaces. While in the first part [Z. Anal. Anwend. 34 (2015), 85-108] optimality
conditions have been derived and analysed in the general case under some assumptions on the quasi-adjoint states, in this second part, we consider diagonal matrices and analyse the corresponding
optimality system without such assumptions.}, author = {Kogut, Peter I. and Kupenko, Ol'Ga P. and Leugering, Günter}, doi = {10.4171/ZAA/1536}, faupublication = {yes}, journal = {Zeitschrift für
Analysis und ihre Anwendungen}, keywords = {Nonlinear monotone Dirichlet problem;control in coefficients;adjoint equation;variable spaces}, month = {Jan}, pages = {199-219}, peerreviewed = {Yes},
title = {{Optimal} {Control} in {Matrix}-{Valued} {Coefficients} for {Nonlinear} {Monotone} {Problems}: {Optimality} {Conditions} {II}}, volume = {34}, year = {2015} } @article{faucris.235775670,
abstract = {This paper is devoted to elliptic fractional boundary value problems of Sturm-Liouville type on an interval and on a general star graph. We first give some existence and uniqueness
results on an open bounded real interval, prove the existence of solutions to a quadratic boundary optimal control problem and provide a characterization via optimality conditions. We then
investigate the analogous problems for a fractional Sturm-Liouville problem on a star graph with mixed Dirichlet and Neumann boundary controls.}, author = {Mophou, Gisele and Leugering, Günter and
Fotsing, Pasquini Soh}, doi = {10.1080/02331934.2020.1730371}, faupublication = {yes}, journal = {Optimization}, note = {CRIS-Team WoS Importer:2020-03-13}, peerreviewed = {Yes}, title = {{Optimal}
control of a fractional {Sturm}-{Liouville} problem on a star graph}, year = {2020} } @article{faucris.239274856, abstract = {We consider a model of population dynamics with age dependence and
spatial structure but unknown birth rate. Using the notion of low-regret [J.-L. Lions, C. R. Acad. ScI. Paris Sér. I Math., 315 (1992), pp. 1253-1257], we prove that we can bring the state of the
system to a desired state by acting on the system via a localized distributed control. We provide the optimality systems that characterize the low-regret control. Moreover, using an appropriate
Hilbert space, we prove that the family of low-regret controls tends to a so-called no-regret control, which we, in turn, characterize.}, author = {Kenne, Cyrille and Leugering, Günter and Mophou,
Gisèle}, doi = {10.1137/19M125875X}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, keywords = {Euler-Lagrange formula; Incomplete data; Low-regret control; No-regret
control; Optimal control; Population dynamics}, note = {CRIS-Team Scopus Importer:2020-06-16}, pages = {1289-1313}, peerreviewed = {Yes}, title = {{Optimal} control of a population dynamics model
with missing birth rate}, volume = {58}, year = {2020} } @article{faucris.107402504, abstract = {In this paper we consider the control of cracks in elastic bodies with rigid inclusion. We first
describe the problem statement, provide an equivalent formulation as a variational inequality and prove existence and uniqueness of solutions. Furthermore, we consider this problem as a limiting
problem when the elasticity parameters of the inclusion tend to infinity. Then we formulate the optimal control problems and derive an explicit formula for the crack sensitivity and for the energy
release rate. We show existence of optimal solutions.}, author = {Khludnev, Alexander M. and Leugering, Günter}, doi = {10.1002/zamm.201000058}, faupublication = {yes}, journal = {ZAMM - Zeitschrift
für angewandte Mathematik und Mechanik}, pages = {125--137}, peerreviewed = {Yes}, title = {{Optimal} control of cracks in elastic bodies with thin rigid inclusions}, volume = {91}, year = {2011} }
@article{faucris.107405584, abstract = {The paper is concerned with the control of the shape of rigid and elastic inclusions and crack paths in elastic bodies. We provide the corresponding problem
formulations and analyze the shape sensitivity of such inclusions and cracks with respect to different perturbations. Inequality type boundary conditions are imposed at the crack faces to provide a
mutual nonpenetration between crack faces. Inclusion and crack shapes are considered as control functions and control objectives, respectively. The cost functional, which is based on the Griffith
rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry. We prove an existence of optimal solutions.}, author = {Khludnev,
A. and Leugering, Günter and Specovius-Neugebauer, M.}, doi = {10.1007/s10957-012-0053-2}, faupublication = {yes}, journal = {Journal of Optimization Theory and Applications}, peerreviewed = {Yes},
title = {{Optimal} {Control} of {Inclusion} and {Crack} {Shapes} in {Elastic} {Bodies}}, year = {2012} } @article{faucris.111266144, author = {Gugat, Martin}, faupublication = {yes}, journal =
{Advanced Modeling and Optimization}, note = {UnivIS-Import:2015-03-09:Pub.2005.nat.dma.zentr.optima}, pages = {9-37}, peerreviewed = {unknown}, title = {{Optimal} {Control} of {Networked}
{Hyperbolic} {Systems}: {Evaluation} of {Derivatives}}, url = {http://www.ici.ro/camo/journal/v7n1.htm}, volume = {7}, year = {2005} } @article{faucris.267819277, abstract = {We study optimal control
problems for time-fractional diffusion equations on metric graphs, where the fractional derivative is considered in the Caputo sense. Using eigenfunction expansions for the spatial part, we first
prove the well-posedness of the system. We then prove the existence of a unique solution to the optimal control problem, where we admit both boundary and distributed controls. We develop an adjoint
calculus for the right Caputo derivative and derive the corresponding first order optimality system. We also propose a finite difference approximation to find the numerical solution of the optimality
system on the graph. In the proposed method, the so-called Ll method is used for the discrete approximation of the Caputo derivative, while the space derivative is approximated using a standard
central difference scheme, which results in converting the optimality system into a system of algebraic equations. Finally, an example is provided to demonstrate the performance of the numerical
method.}, author = {Mehandiratta, Vaibhav and Mehra, Mani and Leugering, Günter}, doi = {10.1137/20M1340332}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, month =
{Jan}, note = {CRIS-Team WoS Importer:2022-01-07}, pages = {4216-4242}, peerreviewed = {Yes}, title = {{OPTIMAL} {CONTROL} {PROBLEMS} {DRIVEN} {BY} {TIME}-{FRACTIONAL} {DIFFUSION} {EQUATIONS} {ON}
{METRIC} {GRAPHS}: {OPTIMALITY} {SYSTEM} {AND} {FINITE} {DIFFERENCE} {APPROXIMATION}}, volume = {59}, year = {2021} } @article{faucris.273567766, abstract = {In the present paper we deal with
parabolic fractional initial-boundary value problems of Sturm-Liouville type in an interval and in a gen-eral star graph. We first give several existence, uniqueness and regularity results of weak
and very-weak solutions. We prove the existence and unique-ness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the
Euler-Lagrange first order optimality conditions. We then investigate the analogous problems for a frac-tional Sturm-Liouville problem in a general star graph with mixed Dirichlet and Neumann
boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange
multipliers.}, author = {Leugering, Günter and Mophou, Gisele and Moutamal, Maryse and Warma, Mahamadi}, doi = {10.3934/mcrf.2022015}, faupublication = {yes}, journal = {Mathematical Control and
Related Fields}, note = {CRIS-Team WoS Importer:2022-04-22}, peerreviewed = {Yes}, title = {{OPTIMAL} {CONTROL} {PROBLEMS} {OF} {PARABOLIC} {FRACTIONAL} {STURM}-{LIOUVILLE} {EQUATIONS} {IN} {A}
{STAR} {GRAPH}}, year = {2022} } @article{faucris.117419984, abstract = {Our goal is to design brittle composite materials yielding maximal energy dissipation for a given static load case. We focus
on the effect of variation of fiber shapes on resulting crack paths and thus on the fracture energy. To this end, we formulate a shape optimization problem, in which the cost function is the fracture
energy and the state problem consists in the determination of the potentially discontinuous displacement field in the two-dimensional domain. Thereby, the behavior at the crack surfaces is modeled by
cohesive laws. We impose a nonpenetration condition to avoid interpenetration of opposite crack sides. Accordingly, the state problem is formulated as variational inequality. This leads to potential
nondifferentiability of the shape-state mapping. For the numerical solution, we derive first-order information in the form of subgradients. We conclude the article by numerical results.}, author =
{Prechtel, Marina and Leugering, Günter and Steinmann, Paul and Stingl, Michael}, doi = {10.1007/s10957-012-0094-6}, faupublication = {yes}, journal = {Journal of Optimization Theory and
Applications}, keywords = {Our goal is to design brittle composite materials yielding maximal energy dissipation for a given static load case. We focus on the effect of variation of fiber shapes on
resulting crack paths and thus on the fracture energy. To this end, we formulate a shape optimization problem, in which the cost function is the fracture energy and the state problem consists in the
determination of the potentially discontinuous displacement field in the two-dimensional domain. Thereby, the behavior at the crack surfaces is modeled by cohesive laws. We impose a nonpenetration
condition to avoid interpenetration of opposite crack sides. Accordingly, the state problem is formulated as variational inequality. This leads to potential nondifferentiability of the shape-state
mapping. For the numerical solution, we derive first-order information in the form of subgradients. We conclude the article by numerical results.}, note =
{UnivIS-Import:2015-03-09:Pub.2011.tech.FT.FT-TM.optima}, pages = {962-985}, peerreviewed = {Yes}, title = {{Optimal} {Design} of {Brittle} {Composite} {Materials}: a {Nonsmooth} {Approach}}, volume
= {155}, year = {2012} } @article{faucris.112053744, abstract = {The Lavrentiev regularization method is a tool to improve the regularity of the Lagrange multipliers in pde constrained optimal
control problems with state constraints. It has already been used for problems with parabolic and elliptic systems. In this paper we consider Lavrentiev regularization for problems with a hyperbolic
system, namely the scalar wave equation. We show that also in this case the regularization yields multipliers in the Hilbert space L^2. We present numerical exam-ples, where we compare the Lavrentiev
regularization, Lavrentiev Prox regularization, a fixed point iteration to improve feasibility, and a penalty method. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.}, author = {Gugat, Martin and
Keimer, Alexander and Leugering, Günter}, doi = {10.1002/zamm.200800196}, faupublication = {yes}, journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik}, keywords = {Hyperbolic
equation; Lavrentiev Prox regularization; Lavrentiev regularization; Optimal control; Pointwise state constraints; Wave equation}, note = {UnivIS-Import:2015-03-09:Pub.2009.nat.dma.zentr.optima},
pages = {420-444}, peerreviewed = {Yes}, title = {{Optimal} distributed control of the wave equation subject to state constraints}, url = {http://www3.interscience.wiley.com/journal/5007542/home},
volume = {89}, year = {2009} } @article{faucris.116675724, abstract = {We consider a finite string where, at both end points, a homogeneous Dirichlet boundary condition holds. One boundary point is
fixed, and the other is moving; hence the length of the string is changing in time. The string is controlled through the movement of this boundary point. We consider movements of the boundary that
are Lipschitz continuous. Only movements for which at the given finite terminal time the string has the same length as at the beginning are admissible. Moreover, we impose an upper bound for the
Lipschitz constant of the movement that is smaller than the speed of wave propagation. We consider the optimal control problem to find an admissible movement for which at the given terminal time the
energy of the string is minimal. We give a sufficient condition for the existence and uniqueness of an optimal movement and construct an optimal control movement. © 2007 Society for Industrial and
Applied Mathematics.}, author = {Gugat, Martin}, doi = {10.1137/06065636x}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, keywords = {Control constraint; Moving
boundary; Optimal boundary control; Optimal control of PDEs; Optimal energy control; Optimal shape; PDE-constrained optimization; Wave equation; 49K20; 35L05}, note =
{UnivIS-Import:2015-03-09:Pub.2007.nat.dma.zentr.optima{\_}0}, pages = {1705-17025}, peerreviewed = {Yes}, title = {{Optimal} {Energy} {Control} in {Finite} {Time} by varying the {Length} of the
{String}}, url = {http://www2.am.uni-erlangen.de/~gugat/sicon2007.pdf}, volume = {46}, year = {2007} } @inproceedings{faucris.200190269, author = {Rathmann, Wigand and Hild, Johannes}, booktitle =
{Proceedings 14. Workshop Mathematik in ingenieurwissenschaftlichen Studiengängen}, date = {2017-09-18/2017-09-18}, editor = {Gottlob-Frege-Zentrum, Hochschule Wismar}, faupublication = {yes}, isbn =
{978-3-942100-55-7}, peerreviewed = {unknown}, title = {{Optimale} {Steuerung} von {Flachwasserkanälen}}, venue = {Friedrich-Alexander-Universität Erlangen-Nürnberg}, year = {2017} } @article
{faucris.110644424, abstract = {The generalized contingent epiderivative of set-valued maps is introduced in this paper and its relationship to the contingent epiderivative is investigated. A unified
necessary and sufficient optimality condition is derived in terms of the generalized contingent epiderivative. The existence of weak subgradients of set-valued maps is proved, and a sufficient
optimality condition of set-valued optimization problems is obtained in terms of weak subgradients.}, author = {Jahn, Johannes and Chen, G.Y.}, faupublication = {yes}, journal = {Mathematical Methods
of Operations Research}, keywords = {Generalized contingent epiderivative; Optimality conditions; Set-valued optimization; Subgradient}, note =
{UnivIS-Import:2015-03-05:Pub.1998.nat.dma.pama21.optima}, pages = {187-200}, peerreviewed = {Yes}, title = {{Optimality} {Conditions} for {Set}-{Valued} {Optimization} {Problems}}, volume = {48},
year = {1998} } @incollection{faucris.118872864, address = {Berlin}, author = {Jahn, Johannes}, booktitle = {Multiple Criteria Decision Making - Proceedings of the Twelfth International Conference,
Hagen (Germany)}, editor = {G. Fandel, T. Gal}, faupublication = {yes}, note = {UnivIS-Import:2015-04-20:Pub.1997.nat.dma.pama21.optima}, pages = {22-30}, peerreviewed = {unknown}, publisher =
{Springer}, series = {Lecture Notes in Economics and Mathematical Systems}, title = {{Optimality} {Conditions} in {Set}-{Valued} {Vector} {Optimization}}, volume = {448}, year = {1997} } @article
{faucris.118529664, abstract = {In this paper we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in L (Ω). In
particular, when the coefficient matrix is taken to satisfy the decomposition B(x) = ρ(x)A(x) with a scalar function ρ, we allow the ρ to degenerate. Such problems are related to various applications
in mechanics, conductivity and to an approach in topology optimization, the SIMP-method. Since equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we
show that the optimal control problem in the coefficients can be stated in different forms depending on the choice of the class of admissible solutions. Using the direct method in the Calculus of
variations, we discuss the solvability of the above optimal control problems in the so-called class of H-admissible solutions. © European Mathematical Society.}, author = {Kogut, Peter I. and
Leugering, Günter}, doi = {10.4171/ZAA/1447}, faupublication = {yes}, journal = {Zeitschrift für Analysis und ihre Anwendungen}, keywords = {Degenerate elliptic equations;control in coefficients;
weighted Sobolev spaces;Lavrentieff phenomenon;direct method in the Calculus of variations}, month = {Jan}, pages = {31--53}, peerreviewed = {Yes}, title = {{Optimal} {L1}-{Control} in {Coefficients}
for {Dirichlet} {Elliptic} {Problems}: {H}-{Optimal} {Solutions}}, volume = {31}, year = {2012} } @article{faucris.107408444, abstract = {In this paper, we study a Dirichlet optimal control problem
associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The coefficients may degenerate and, therefore, the problems may exhibit
the so-called Lavrentieff phenomenon and non-uniqueness of weak solutions. We consider the solvability of this problem in the class of W-variational solutions. Using a concept of variational
convergence of constrained minimization problems in variable spaces, we prove the existence of W-solutions to the optimal control problem and provide the way for their approximation. We emphasize
that control problems of this type are important in material and topology optimization as well as in damage or life-cycle optimization.}, author = {Kogut, Peter I. and Leugering, Günter}, doi =
{10.1007/s10957-011-9840-4}, faupublication = {yes}, journal = {Journal of Optimization Theory and Applications}, keywords = {Degenerate elliptic equations;Control in coefficients;Weighted Sobolev
spaces;Lavrentieff phenomeno;Variational convergence}, pages = {205--232}, peerreviewed = {Yes}, title = {{Optimal} {L1}-{Control} in {Coefficients} for {Dirichlet} {Elliptic} {Problems}: {W}-
{Optimal} {Solutions}}, volume = {150}, year = {2011} } @misc{faucris.119422204, abstract = {
The number of dual-career couples with children is growing fast. These
couples face various challenging problems of organizing their lifes, in par-
ticular connected with childcare and time-management. As a typical ex-
ample we study one of the difficult decision problems of a dual career
couple from the point of view of operations research with a particular
focus on gender equality, namely the location problem to find a family
home. This leads to techniques that allow to include the value of gender
equality in rational decision processes.
}, author = {Gugat, Martin and Abele-Brehm, Andrea and Klamroth, Kathrin}, faupublication = {yes}, keywords = {Dual Careec couples, Optimal Location Problem, Childcare Management}, peerreviewed =
{automatic}, title = {{Optimal} location of family homes for dual career couples}, url = {http://www.optimization-online.org/DB{\_}HTML/2010/02/2547.html}, year = {2010} } @article{faucris.117325824,
abstract = {
We consider a vibrating string that is fixed at one end with Neumann control action at the other end. We investigate the optimal control problem of steering this system from given initial data to
rest, in time T, by minimizing an objective functional that is the convex sum of the L^2-norm of the control and of a boundary Neumann tracking term.
We provide an explicit solution of this optimal control problem, showing that if the weight of the tracking term is positive, then the optimal control action is concentrated at the beginning and at
the end of the time interval, and in-between it decays exponentially. We show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an
exponentially increasing term. This implies that, if the time T is large, then the optimal trajectory approximately consists of three arcs, where the first and the third short-time arcs are transient
arcs, and in the middle arc the optimal control and the corresponding state are exponentially close to 0. This is an example of a turnpike phenomenon for a problem of optimal boundary control. If T=
+∞ (infinite time horizon problem), then only the exponentially decaying component of the control remains, and the norms of the optimal control action and of the optimal state decay exponentially in
time. In contrast to this situation, if the weight of the tracking term is zero and only the control cost is minimized, then the optimal control is distributed uniformly along the whole interval
[0,T] and coincides with the control given by the Hilbert Uniqueness Method.
In addition, we establish a similarity theorem stating that, for every T>0, there exists an appropriate weight λ<1 for which the optimal solutions of the corresponding finite horizon optimal control
problem and of the infinite horizon optimal control problem coincide along the first part of the time interval [0,2]. We also discuss the turnpike phenomenon from the perspective of a general
framework with a strongly continuous semi-group.
}, author = {Gugat, Martin and Trelat, Emmanuel and Zuazua, Enrique}, doi = {10.1016/j.sysconle.2016.02.001}, faupublication = {yes}, journal = {Systems & Control Letters}, keywords = {Vibrating
string; Neumann boundary control; Turnpike phenomenon; Exponential stability; Energy decay; Exact control; Infinite horizon optimal control; Similarity theorem; Receding horizon}, pages = {61-70},
peerreviewed = {Yes}, title = {{Optimal} {Neumann} control for the {1D} wave equation: {Finite} horizon, infinite horizon, boundary tracking terms and the turnpike property}, volume = {90}, year =
{2016} } @article{faucris.111889624, abstract = {In many control application, switching between different control devices occurs. Here the problem to control a finite string to the zero state in
finite time by controlling the state at the two boundary points is considered, where at each moment in time one of the boundary controls must be switched off, that is its control value must be equal
to zero. The corresponding optimal control problem where the objective function is the L^2 norm of the controls is solved explicitly in the sense that controls that are successful and minimize at the
same time the objective function are determined as functions of the initial state. Due to the complementarity condition that appears in the optimal control problem, it is non-convex and the optimal
control is in general not uniquely determined. To allow for technical constraints it is important to avoid an accumulation of switching points at so-called Zeno-points. We give examples that
illustrate how switching regimes of practical value can be obtained. © 2008 WILEY-VCH Verlag GmbH & Co. KGaA.}, author = {Gugat, Martin}, doi = {10.1002/zamm.200700154}, faupublication = {yes},
journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik}, keywords = {MPEC; Optimal boundary control; PDE constrained optimization; Switching control; Wave equation}, note =
{UnivIS-Import:2015-03-09:Pub.2008.nat.dma.zentr.optima}, pages = {283-305}, peerreviewed = {Yes}, title = {{Optimal} switching boundary control of a string to rest in finite time}, url = {http://
www3.interscience.wiley.com/cgi-bin/abstract/117946700/ABSTRACT}, volume = {88}, year = {2008} } @article{faucris.106688824, author = {Rüffler, Fabian and Hante, Falk}, doi = {10.1016/
j.nahs.2016.05.001}, faupublication = {yes}, journal = {Nonlinear Analysis: Hybrid Systems}, keywords = {Partial differential equation }, pages = {215 - 227}, peerreviewed = {unknown}, title =
{{Optimal} switching for hybrid semilinear evolutions}, url = {http://www.sciencedirect.com/science/article/pii/S1751570X16300231}, volume = {22}, year = {2016} } @article{faucris.117348924, author =
{Eichfelder, Gabriele and Gugat, Martin}, faupublication = {yes}, journal = {Das Wirtschaftsstudium}, note = {UnivIS-Import:2015-03-09:Pub.2010.nat.dma.zentr.optimi}, pages = {571-578}, peerreviewed
= {No}, title = {{Optimierung} mit mehreren konkurrierenden {Zielen}}, year = {2010} } @inproceedings{faucris.276080478, address = {NEW YORK}, author = {Artola, Marc and Rodriguez, Charlotte and
Wynn, Andrew and Palacios, Rafael and Leugering, Günter}, booktitle = {2021 60TH IEEE CONFERENCE ON DECISION AND CONTROL (CDC)}, doi = {10.1109/CDC45484.2021.9683680}, faupublication = {yes}, month =
{Jan}, note = {CRIS-Team WoS Importer:2022-05-27}, pages = {6043-6048}, peerreviewed = {unknown}, publisher = {IEEE}, title = {{Optimisation} of {Region} of {Attraction} {Estimates} for the
{Exponential} {Stabilisation} of the {Intrinsic} {Geometrically} {Exact} {Beam} {Model}}, venue = {, ELECTR NETWORK}, year = {2021} } @incollection{faucris.117209224, abstract = {We present topology
optimization of piezoelectric loudspeakers using the SIMP method and topology gradient based methods along with analytical and numerical results.}, address = {Basel}, author = {Bänsch, Eberhard and
Kaltenbacher, Manfred and Leugering, Günter and Schury, Fabian and Wein, Fabian}, booktitle = {Constrained Optimization and Optimal Control for Partial Differential Equations}, doi = {10.1007/
978-3-0348-0133-1{\_}26}, editor = {Günter Leugering, Sebastian Engell, Andreas Griewank, Michael Hinze, Rolf Rannacher, Volker Schulz, Michael Ulbrich, Stefan Ulbrich,}, faupublication = {yes}, isbn
= {978-3-0348-0132-4}, keywords = {Topology optimization; piezoelectricity; loudspeakers}, pages = {501-519}, peerreviewed = {unknown}, publisher = {Birkhäuser/ Springer Basel AG}, series =
{International Series of Numerical Mathematics}, title = {{Optimization} of electro-mechanical smart structures}, volume = {160}, year = {2012} } @article{faucris.111256684, abstract = {Two types of
rod antennas of mobile phones are optimized so that the radiated energy absorbed by the head or body of the user is reduced and the radiation intensity to other areas especially to the receiver is
increased. The mathematical modelling of this problem leads to an infinite dimensional bicriterial optimization problem. It is shown that this optimization problem and a discretized version of this
problem are solvable. The relationship between the infinite and finite dimensional optimization problem is investigated. Numerical results are presented for mobile phones working with the GSM
standards 900 and 1800. © Springer-Verlag 2004.}, author = {Jahn, Johannes and Kirsch, A. and Wagner, C.}, doi = {10.1007/s001860300318}, faupublication = {yes}, journal = {Mathematical Methods of
Operations Research}, keywords = {Antenna theory; Multiobjective optimization}, note = {UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pama21.optimi}, pages = {37-51}, peerreviewed = {Yes}, title =
{{Optimization} of {Rod} {Antennas} of {Mobile} {Phones}}, volume = {59}, year = {2004} } @article{faucris.121714164, abstract = {Resonance frequency analysis (RFA) using the Osstell device (Osstell
AB, Gothenburg, Sweden) has been advocated for quantifying implant stability on a relative scale of implant stability quotients (ISQ). It was the goal of this prospective clinical study to evaluate
whether a certain ISQ level, at the time an implant is placed, correlates with successful osseointegration as some have claimed. Four hundred ninety-five implants (Straumann AG, Basel, Switzerland),
varying in length and diameter, were placed in a private practice, strictly adhering to the implant manufacturer's surgical protocol. After placement and after healing periods of 42 days in the
mandible and 56 days (implant manufacturer's protocol) in the maxilla, implant stability was measured using RFA. After healing, implants were torqued forward at 35 Ncm and allowed to heal further if
the patients felt discomfort. Statistical analysis of the data obtained was based on Welch tests and Kolmogorov-Smirnow tests (level of significance ? = 0.05). Results showed that 432 implants were
osseointegrated after the predefined healing periods while 8 implants were lost and, in 55 cases, healing was prolonged. Both at insertion (P = .025) and after healing (P < .001), successful implants
showed significantly different ISQ values as compared to implant failures or implants with prolonged healing. However, overlapping ISQ distributions at implant insertion demonstrated that there was
no correlation among the data that could be used to predict successful osseointegration. Within the limits of this study, the prognostic value of ISQ values appears to be ambiguous.}, author =
{Krafft, Tim and Graef, Friedrich and Karl, Matthias}, doi = {10.1563/AAID-JOI-D-13-00172}, faupublication = {yes}, journal = {The Journal of oral implantology}, note = {EVALuna2:24823}, pages =
{e133-7}, peerreviewed = {Yes}, title = {{Osstell} {Resonance} {Frequency} {Measurement} {Values} as a {Prognostic} {Factor} in {Implant} {Dentistry}}, volume = {41}, year = {2015} } @article
{faucris.119206604, abstract = {The visual appearance of the artificial world is largely governed by films or composites containing particles with at least one dimension smaller than a micron. Over
the past century and a half, the optical properties of such materials have been scrutinized and a broad range of colorant products, based mostly on empirical microstructural improvements, developed.
With the advent of advanced synthetic approaches capable of tailoring particle shape, size and composition on the nanoscale, the question of what is the optimum particle for a certain optical
property can no longer be answered solely by experimentation. Instead, new and improved computational approaches are required to invert the structure-function relationship. This progress report
reviews the development in our understanding of this relationship and indicates recent examples of how theoretical design is taking an ever increasingly important role in the search for enhanced or
multifunctional colorants. Copyright © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei}, author = {Klupp Taylor, Robin and Seifrt, Franti ek and Zhuromskyy, Oleksandr and Peschel, Ulf and Leugering,
Günter and Peukert, Wolfgang}, doi = {10.1002/adma.201100541}, faupublication = {yes}, journal = {Advanced Materials}, keywords = {colloids; core/shell nanoparticles; designed nanostructures;
functional coatings; structure-property relationships}, pages = {2554-2570}, peerreviewed = {Yes}, title = {{Painting} by numbers: {Nanoparticle}-based colorants in the post-empirical age}, volume =
{23}, year = {2011} } @inproceedings{faucris.114001184, author = {Rathmann, Wigand}, booktitle = {4. SAXON SIMULATION MEETING and Mathcad-Workshop}, date = {2012-04-16/2012-04-17}, faupublication =
{yes}, peerreviewed = {No}, publisher = {TU Chemnitz, Professur für Montage- und Handhabungstechnik}, title = {{Parameterschätzung} in gewöhnlichen {Differentialgleichungen}}, url = {http://
nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-84901}, year = {2012} } @article{faucris.110301884, abstract = {In this paper, a known scalarization result of vector optimization theory is reviewed and
stated in a different form and a new short proof is presented. Moreover, it is shown how to apply this result to multi-objective optimization problems and to special problems in statistics and
optimal control theory. © 1987 Plenum Publishing Corporation.}, author = {Jahn, Johannes}, doi = {10.1007/BF00940199}, faupublication = {yes}, journal = {Journal of Optimization Theory and
Applications}, keywords = {control approximation; covariance matrices; Multi-objective optimization}, note = {UnivIS-Import:2015-03-05:Pub.1987.nat.dma.pama21.parame}, pages = {503-516}, peerreviewed
= {Yes}, title = {{Parametric} {Approximation} {Problems} {Arising} in {Vector} {Optimization}}, volume = {54}, year = {1987} } @inproceedings{faucris.121757944, address = {Turnhout}, author =
{Gugat, Martin}, booktitle = {Karl der Große und sein Nachwirken: 1200 Jahre Kultur und Wissenschaft in Europa. Band 2: Mathematisches Wissen}, date = {1995-03-19/1995-03-26}, faupublication = {no},
isbn = {2-503-50674-7/hbk}, keywords = {parametric optimization; Lagrange function}, note = {UnivIS-Import:2015-04-16:Pub.1998.nat.dma.lama1.parame}, pages = {471-483}, peerreviewed = {Yes},
publisher = {Brepols}, title = {{Parametric} {Convex} {Optimization}: {One}-{Sided} {Derivatives} of the {Value} {Function} in {Singular} {Parameters}}, venue = {Aachen}, year = {1998} } @article
{faucris.110561484, abstract = {We consider a countable family of one-parameter convex programs and give sufficient conditions for the one-sided differentiability of its optimal value function. The
analysis is based on the Borwein dual problem for a family of convex programs (a convex disjunctive program). We give conditions that assure stability of the situation of perfect duality in the
Borwein theory. For the reader's convenience, we start with a review of duality results for families of convex programs. A parametric family of dual problems is introduced that contains the dual
problems of Balas and Borwein as special cases. In addition, a vector optimization problem is defined as a dual problem. This generalizes a result by Helbig about families of linear programs.},
author = {Gugat, Martin}, faupublication = {no}, journal = {Journal of Optimization Theory and Applications}, keywords = {Disjunctive programming; Efficiency; Families of convex programs; Mixed
integer programming; Optimal value function; Parameteric family of dual problems; Sensitivity}, note = {UnivIS-Import:2015-03-05:Pub.1997.nat.dma.lama1.parame}, pages = {285-310}, peerreviewed =
{Yes}, title = {{Parametric} {Disjunctive} {Programming}: {One}-{Sided} {Differentiability} of the {Value} {Function}}, volume = {92}, year = {1997} } @incollection{faucris.110515504, address =
{Oberwolfach}, author = {Stingl, Michael and Schmidt, Bastian}, booktitle = {Mini-Workshop: Geometries, Shapes and Topologies in PDE-based Applications}, doi = {10.4171/OWR/2012/57}, editor =
{Michael Hintermüller, Günter Leugering. Jan Sokolowski}, faupublication = {yes}, pages = {3400-3403}, peerreviewed = {unknown}, publisher = {European Mathematical Society Publishing House}, series =
{Oberwolfach Reports}, title = {{Parametric} {Shape} {Optimization} {Revisited}}, volume = {57}, year = {2012} } @article{faucris.207256326, abstract = {The haze factor, which describes the fraction
of light that is scattered when passing through a transparent material, is of general importance for any optical device, from milk glass shielding visibility while providing ambient lighting to solar
cells that are optimized by sophisticated light management layers. Often, such active layers are fabricated from particulate materials that are deposited as thin films on a substrate. Here, the
effect of structural arrangement, position, and orientation of particles on the resulting haze factor is investigated. A mathematical optimization model that iteratively alters the particle layer
structure to maximize or minimize the haze factor for a range of optimization scenarios is designed. Colloidal self-assembly techniques are then used to replicate typical particle structures found in
the optimized designs and correlate the macroscopically measured haze values to the predictions of the optimization. The results indicate general design rules that control the haze value in particle
layers. Non close-packed structures with distributed scatterers and high degrees of order provide minimal haze values while chain-like arrangements and small clusters maximize the haze of a particle
layer. Finally, the findings are transferred to metal nanohole films as model transparent electrodes with controlled haze values.
}, author = {Semmler, Johannes and Bley, Karina and Klupp Taylor, Robin and Stingl, Michael and Vogel, Nicolas}, doi = {10.1002/adfm.201806025}, faupublication = {yes}, journal = {Advanced Functional
Materials}, peerreviewed = {unknown}, title = {{Particulate} {Coatings} with {Optimized} {Haze} {Properties}}, url = {https://onlinelibrary.wiley.com/doi/full/10.1002/adfm.201806025}, volume = {5},
year = {2018} } @book{faucris.109705244, abstract = {A class of nonsmooth shape optimization problems for variational inequalities is considered. The variational inequalities model elliptic boundary
value problems with the Signorini type unilateral boundary conditions. The shape functionals are given by the first order shape derivatives of the elastic energy. In such a way the singularities of
weak solutions to elliptic boundary value problems can be characterized. An example in solid mechanics is given by the Griffith’s functional, which is defined in plane elasticity to measure SIF, the
so-called stress intensity factor, at the crack tips. Thus, topological optimization can be used for passive control of singularities of weak solutions to variational inequalities. The Hadamard
directional differentiability of metric the projection onto the positive cone in fractional Sobolev spaces is employed to the topological sensitivity analysis of weak solutions of nonlinear elliptic
boundary value problems. The first order shape derivatives of energy functionals in the direction of specific velocity fields depend on the solutions to variational inequalities in a subdomain. A
domain decomposition technique is used in order to separate the unilateral boundary conditions and the energy asymptotic analysis. The topological derivatives of nonsmooth integral shape functionals
for variational inequalities are derived. Singular geometrical domain perturbations in an elastic body Ω are approximated by regular perturbations of bilinear forms in variational inequality, without
any loss of precision for the purposes of the secondorder shape-topological sensitivity analysis. The second-order shape-topological directional derivatives are obtained for the Laplacian and for
linear elasticity in two and three spatial dimensions. In the proposed method of sensitivity analysis, the singular geometrical perturbations ϵ→ω ⊂ Ω centred at x ∈ Ω are replaced by regular
perturbations of bilinear forms supported on the manifold Γ ={|x−x|=R} in an elastic body, with R > ϵ > 0. The obtained expressions for topological derivatives are easy to compute and therefore
useful in numerical methods of topological optimization for contact problems.}, author = {Leugering, Günter and Sokolowski, Jan and Zochowski, Antoni}, doi = {10.1007/978-3-319-30785-5{\_}4},
faupublication = {yes}, pages = {65-102}, peerreviewed = {unknown}, publisher = {Springer International Publishing}, title = {{Passive} control of singularities by topological optimization: {The}
second-order mixed shape derivatives of energy functionals for variational inequalities}, volume = {109}, year = {2016} } @article{faucris.107372804, abstract = {In these notes we consider
PDE-constrained optimization in the context of advanced materials. We give examples for optimization and control in the coefficients, free material optimization, topology optimization, shape
optimization in elastic and piezo-electric materials. We explain some approaches to metamaterials, in particular for auxetic materials}, author = {Leugering, Günter and Stingl, Michael}, doi =
{10.1002/gamm.201010016}, faupublication = {yes}, journal = {GAMM-Mitteilungen}, keywords = {piezoelectricity; electromechanical interaction; shape sensitivity analysis}, pages = {209-229},
peerreviewed = {unknown}, title = {{PDE}-constrained optimization for advanced materials}, url = {http://onlinelibrary.wiley.com/doi/10.1002/gamm.201010016/epdf}, volume = {33}, year = {2010} }
@article{faucris.121354464, author = {Gugat, Martin}, doi = {10.1137/080725921}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, keywords = {wave equation, Tychonov
regularization, PDE constrained optimization, penal- ization, exact penalty, exact controllability, state constraints, smoothed exact penalty, speed of convergence, hyperbolic PDE, optimal control},
note = {UnivIS-Import:2015-03-09:Pub.2009.nat.dma.zentr.penalt}, pages = {3026-3051}, peerreviewed = {Yes}, title = {{Penalty} {Techniques} for {State} {Constrained} {Optimal} {Control} {Problems}
with the {Wave} {Equation}}, url = {http://epubs.siam.org/doi/abs/10.1137/080725921}, volume = {48}, year = {2009} } @incollection{faucris.123288264, abstract = {We compare the quality and generation
performance of the optimal control sequence produced by the software frameworks BlueM.MPC and Lamatto.}, author = {Heusch, Steffen and Hild, Johannes and Leugering, Günter and Ostrowski, Manfred},
booktitle = {Mathematical Optimization of Water Networks}, doi = {10.1007/978-3-0348-0436-3{\_}9}, editor = {Birkhäuser Basel}, faupublication = {yes}, isbn = {978-3-0348-0435-6}, pages = {151-165},
peerreviewed = {unknown}, publisher = {Springer Basel}, series = {International Series of Numerical Mathematics}, title = {{Performance} and {Comparison} of {BlueM}.{MPC} and {Lamatto}}, volume =
{162}, year = {2012} } @incollection{faucris.120448064, author = {Jahn, Johannes}, booktitle = {Methods of Operations Research}, faupublication = {yes}, note =
{UnivIS-Import:2015-03-05:Pub.1987.nat.dma.pama21.proble}, pages = {51-59}, peerreviewed = {unknown}, title = {{Problems} of {Vector} {Optimization}}, volume = {57}, year = {1987} } @article
{faucris.118517124, abstract = {In this contribution the identification of new reaction conditions for the production of nearly monodisperse silicon nanoparticles via the pyrolysis of monosilane in a
hot wall reactor is considered. For this purpose a full finite volume model has been combined with a state-of-the-art trust-region optimisation algorithm for process control. Verified against
experimental data, specific process conditions are determined accomplishing a versatile range of prescribed product properties. The main achievement of the optimisation is the possibility to control
the different mechanisms in the particle formation process by mainly adjusting the temperature profile. Due to a successful separation of the nucleation and growth process, significantly narrower
particle size distributions are obtained. Moreover, the presented optimisation framework establishes rate constants based on measured data.}, author = {Gröschel, Michael and Koermer, Richard and
Walther, Maximilian and Leugering, Günter and Peukert, Wolfgang}, doi = {10.1016/j.ces.2012.01.035}, faupublication = {yes}, journal = {Chemical Engineering Science}, pages = {181--194}, peerreviewed
= {Yes}, title = {{Process} control strategies for the gas phase synthesis of silicon nanoparticles}, volume = {73}, year = {2012} } @article{faucris.222304913, author = {Geiger, Manfred and Meßner,
Arthur and Engel, Ulf}, faupublication = {yes}, journal = {Production Engineering}, note = {LFT Import::2019-07-15 (650)}, peerreviewed = {Yes}, title = {{Production} of microparts - size effects in
bulk metal {Forming}, similarity theory.}, year = {1996} } @incollection{faucris.115535684, address = {Rocquencourt}, author = {Jahn, Johannes}, booktitle = {Third Franco-German Conference in
Optimization}, faupublication = {no}, note = {UnivIS-Import:2015-04-17:Pub.1985.nat.dma.pama21.proper}, pages = {27-28}, peerreviewed = {unknown}, publisher = {INRIA Report}, title = {{Properly}
{Minimal} {Elements} of a {Set}}, year = {1985} } @article{faucris.119588304, author = {Truong, Xuan Duc Ha and Jahn, Johannes}, faupublication = {yes}, journal = {Journal of Nonlinear and Convex
Analysis}, pages = {415 - 429}, peerreviewed = {Yes}, title = {{Properties} of {Bishop}-{Phelps} cones}, volume = {18}, year = {2017} } @article{faucris.110637824, abstract = {If a fractional program
does not have a unique solution or the feasible set is unbounded, numerical difficulties can occur. By using a prox-regularization method that generates a sequence of auxiliary problems with unique
solutions, these difficulties are avoided. Two regularization methods are introduced here. They are based on Dinkelbach-type algorithms for generalized fractional programming, but use a regularized
parametric auxiliary problem. Convergence results and numerical examples are presented.}, author = {Gugat, Martin}, doi = {10.1023/A:1021759318653}, faupublication = {no}, journal = {Journal of
Optimization Theory and Applications}, keywords = {differential correction method; Dinkelbach algorithm; Generalized fractional programs; ill-posed problems; linear convergence; prox-regularization},
note = {UnivIS-Import:2015-03-05:Pub.1998.nat.dma.lama1.proxre}, pages = {691-722}, peerreviewed = {Yes}, title = {{Prox}-{Regularization} {Methods} for {Generalized} {Fractional} {Programming}}, url
= {http://link.springer.com/article/10.1023/A:1021759318653}, volume = {99}, year = {1998} } @phdthesis{faucris.106824124, abstract = {We study the shallow water equations augmented with pollution
transport in one space dimension on prismatic channels with nontrivial channel width. We use both, the characteristic and the weak formulation, to construct an explicit solution of the corresponding
Riemann problem. This solution is an important ingredient for Godunov's Finite volume scheme, which is used to discretize a real world sewer system into a network of finite volume elements. We
especially focus on a finite volume model for the network vertices, which implies weak coupling conditions at each vertex and is based on the solution of a transjunctional Riemann problem. In order
to develop a process model for sewer systems of practical relevance, we introduce generalized numerical flux functions, which allow the modeling of wave reflections at nonprismatic channel junctions
and walls. The generalized numerical flux functions also enable the modeling of controllable special structures like pumps, weirs and valves. The resulting process model is especially suited for real
world sewer systems and is mathematically represented by a system of ordinary differential equations. The controllable process model is subject to an optimal control problem for real-time
application. We discretize the process model with a general linear one-step scheme in time and use adjoint calculus to provide explicit formulas for the gradient and Hessian of the time discrete cost
functional. We verify our approach with a self-written C++ application, which is tailored to optimize finite volume networks in real-time, and provide numerical results for two sewer systems, which
are both based on practical application. We apply a receding horizon strategy to both test cases and compare the real-time control results with the previously computed solutions of an online control
approach.}, author = {Hild, Johannes}, faupublication = {yes}, school = {Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{Real}-{Time} {Control} of {Hydrodynamic} {Process} {Models} on
{Finite} {Volume} {Networks}}, year = {2012} } @incollection{faucris.122351724, abstract = {A hydrodynamic process model based on shallow water equations is discretized on 1D-networks with the method
of finite volumes. Based on the finite volumes we replace algebraic coupling conditions by a consistent finite volume junction model. We use discrete adjoint computation for one step Runge-Kutta
schemes to generate fast and robust gradients for descent methods. We use the descent methods to generate an optimal control for an example network and discuss the computational results.}, author =
{Hild, Johannes and Leugering, Günter}, booktitle = {Mathematical Optimization of Water Networks}, doi = {10.1007/978-3-0348-0436-3{\_}8}, editor = {Birkhäuser Basel}, faupublication = {yes}, isbn =
{978-3-0348-0435-6}, pages = {pp 129-150}, peerreviewed = {unknown}, publisher = {Springer Basel}, series = {International Series of Numerical Mathematics}, title = {{Real}-{Time} {Control} of
{Urban} {Drainage} {Systems}}, volume = {162}, year = {2012} } @book{faucris.109061964, address = {Berlin}, editor = {Jahn, Johannes and Krabs, W.}, faupublication = {yes}, note =
{UnivIS-Import:2015-05-08:Pub.1987.nat.dma.pama21.dualit}, publisher = {Springer}, title = {{Recent} {Advances} and {Historical} {Development} of {Vector} {Optimization}}, year = {1987} } @article
{faucris.115216464, abstract = {This paper presents a reference point approximation algorithm which can be used for the interactive solution of bicriterial nonlinear optimization problems with
inequality and equality constraints. The advantage of this method is that the decision maker may choose arbitrary reference points in the criteria space. Moreover, a special tunneling technique is
given for the computation of global solutions of certain subproblems. Finally, the proposed method is applied to a mathematical example and a problem in mechanical engineering. © 1992 Plenum
Publishing Corporation.}, author = {Jahn, Johannes and Merkel, A.}, doi = {10.1007/BF00939894}, faupublication = {yes}, journal = {Journal of Optimization Theory and Applications}, keywords =
{interactive methods; Multi-objective optimization}, note = {UnivIS-Import:2015-03-05:Pub.1992.nat.dma.pama21.refere}, pages = {87-103}, peerreviewed = {Yes}, title = {{Reference} {Point}
{Approximation} {Method} for the {Solution} of {Bicriterial} {Nonlinear} {Optimization} {Problems}}, volume = {74}, year = {1992} } @article{faucris.117916084, abstract = {We present an algorithm
that automatically registers one segmented contour in a frame of the video sequence to the contour in the next frame to derive discrete 2-D trajectories of PE vibrations. By concatenation of the
obtained transformations this approach provides a total registration of PE segment contours. We suggest a mixed-integer programming formulation for the problem that combines an advanced outlier and
deformation handling with the introduction of dummy points in regions that newly open up, and that includes normal information in the objective function to avoid unwanted deformations. Numerical
experiments show that the implemented alternate convex search algorithm produces robust results which is demonstrated at the example of five high-speed recordings of laryngectomee subjects. (C) 2007
Elsevier B.V. All rights reserved.}, author = {Stiglmayr, Michael and Schwarz, Raphael and Klamroth, Kathrin and Leugering, Günter and Lohscheller, Jörg}, doi = {10.1016/j.media.2007.12.001},
faupublication = {yes}, journal = {Medical Image Analysis}, keywords = {substitute voice;PE segment;digital high-speed videos;point matching;shape registration;generalized assignment problem;2-D
trajectories}, pages = {318-334}, peerreviewed = {Yes}, title = {{Registration} of {PE} segment contour deformations in digital high-speed videos}, url = {http://www.sciencedirect.com/science/article
/pii/S1361841507001120}, volume = {12}, year = {2008} } @article{faucris.107398324, abstract = {In this contribution the optimal boundary control problem for a first order nonlinear, nonlocal
hyperbolic PDE is studied. Motivated by various applications ranging from re-entrant manufacturing systems to particle synthesis processes, we establish the regularity of solutions for W1,p-data.
Based on a general L2 tracking type cost functional, the existence, uniqueness, and regularity of the adjoint system in W1,p is derived using the special structure induced from the nonlocal flux
function of the state equation. The assumption of W1,p - and not Lp-regularity comes thereby due to the fact that the adjoint equation asks for more regularity to be well defined. This problem is
discussed in detail, and we give a solution by defining a special type of cost functional, such that the corresponding optimality system is well defined.}, author = {Gröschel, Michael and Keimer,
Alexander and Leugering, Günter and Wang, Zhiqiang}, doi = {10.1137/120873832}, faupublication = {yes}, journal = {SIAM Journal on Control and Optimization}, pages = {2141--2163}, peerreviewed =
{Yes}, title = {{Regularity} {Theory} and {Adjoint}-{Based} {Optimality} {Conditions} for a {Nonlinear} {Transport} {Equation} with {Nonlocal} {Velocity}}, volume = {52}, year = {2014} } @article
{faucris.110992904, abstract = {
We consider L ^∞-norm minimal controllability problems for vibrating systems. In the common method of modal truncation controllability constraints are first reformulated as an infinite sequence of
moment equations, which is then truncated to a finite set of equations. Thus, feasible controls are represented as solutions of moment problems.
In this paper, we propose a different approach, namely to replace the sequence of moment equations by a sequence of moment inequalities. In this way, the feasible set is enlarged. If a certain
relaxation parameter tends to zero, the enlarged sets approach the original feasible set. Numerical examples illustrate the advantages of this new approach compared with the classical method of
The introduction of moment inequalities can be seen as a regularization method, that can be used to avoid oscillatory effects. This regularizing effect follows from the fact that for each relaxation
parameter, the whole sequence of eigenfrequencies is taken into account, whereas in the method of modal truncation, only a finite number of frequencies is considered.
}, author = {Gugat, Martin and Leugering, Günter}, doi = {10.1023/A:1015472323967}, faupublication = {no}, journal = {Computational Optimization and Applications}, keywords = {optimal control; exact
controllability; eigenvalues; moment problem; moment inequalities; numerical algorithm; convergence}, note = {UnivIS-Import:2015-03-09:Pub.2002.nat.dma.lama1.regula}, pages = {151-192}, peerreviewed
= {Yes}, title = {{Regularization} of {L}-∞ optimal control problems for distributed parameter systems}, url = {http://link.springer.com/article/10.1023/A:1015472323967}, volume = {22}, year = {2002}
} @article{faucris.109705684, abstract = {This paper concerns the generalization and regularization of a nonlinear scalarization method by Pascoletti and Serafini, [14], to abstract partially ordered
Banach spaces. In particular, we put emphasis on the possibility that the ordering cone might have an empty interior. All this is motivated by applying the method to a vector optimization problem
with PDE-constraints. © 2012 WILEY-VCH Verlag GmbH & Co. KGaA.}, author = {Leugering, Günter and Schiel, Ralph}, doi = {10.1002/gamm.201210014}, faupublication = {yes}, journal = {GAMM-Mitteilungen},
keywords = {Gerstewitz function; Nonlinear scalarization; Nonsolid ordering cones; Optimal control; Vector optimization}, pages = {209-225}, peerreviewed = {Yes}, title = {{Regularized} nonlinear
scalarization for vector optimization problems with {PDE}-constraints}, volume = {35}, year = {2012} } @article{faucris.123654564, author = {Hante, Falk and Sager, Sebastian}, doi = {10.1007/
s10589-012-9518-3}, faupublication = {yes}, journal = {Computational Optimization and Applications}, pages = {197--225}, peerreviewed = {Yes}, title = {{Relaxation} methods for mixed-integer optimal
control of partial differential equations}, volume = {55}, year = {2013} } @inproceedings{faucris.120852424, abstract = {In this paper we consider some classes of discrete-continuous 2-D models in
order to build a finite dimensional linear state space model of a gas distribution network. The models introduced are shown to be suitable for handling problems of optimal control of pressure and
flow in gas transport units. The focus is on the development of a comprehensive optimization theory based on a 'constructive approach'. Moreover, an optimal feedback control problem is considered
that is of interest in both systems theory and application.}, author = {Dymkou, Siarhei and Leugering, Günter and Jank, Gerhard}, booktitle = {Multidimensional (nD) Systems}, date = {2007-06-27/
2007-06-29}, doi = {10.1109/NDS.2007.4509556}, editor = {IEEE}, faupublication = {yes}, keywords = {Pipelines; Mathematical model; State-space methods; Optimal control; Transportation; Large-scale
systems; Optimization methods; Control theory; Fluid flow; Isothermal processes}, month = {Jan}, pages = {101-108}, peerreviewed = {unknown}, publisher = {IEEE Xplore}, title = {{Repetitive}
processes modelling of gas transport networks}, url = {http://ieeexplore.ieee.org/document/4509556/}, year = {2007} } @inproceedings{faucris.122844744, author = {Gugat, Martin and Tucsnak, Marius and
Sigalotti, Mario}, booktitle = {Proceedings of the 9th European Control Conference, Kos, Greece, 2007.}, date = {2007-07-02/2007-07-05}, doi = {10.23919/ECC.2007.7068766}, faupublication = {yes},
isbn = {978-3-9524173-8-6}, keywords = {vibrating string; boundary control; robustness; position term;}, note = {UnivIS-Import:2015-04-16:Pub.2007.nat.dma.zentr.robust}, pages = {-}, publisher =
{IEEE}, title = {{Robustness} analysis for the boundary control of the string equation}, url = {https://ieeexplore.ieee.org/document/7068766}, venue = {Kos}, year = {2007} } @inproceedings
{faucris.235431970, abstract = {
Hands-on Tutorial
Für ein besseres Verständnis helfen interaktive Inhalte weiter. Im Bereich der Statistik oder der Darstellung von dreidimensionalen Objekten fehlte bisher die Möglichkeit, dies direkt in Lernmodulen
einzubinden. Mit dem neuen Content-Element "SageMath-Cell" kann nun z.B. direkt R-Code in Lernmodulen genutzt werden. SageMath ist ein mathematisches Softwaresystem, das die Nutzung vieler
Open-Source-Pakete, wie Maxima oder R, zusammenfasst; es kann auch Python genutzt werden. In diesem Workshop wird an Beispielen erprobt, wie das Content-Element aus Autorensicht angewendet werden
kann. Ihre Ideen können Sie gerne mitbringen – und bitte auch Ihren Compute}, author = {Rathmann, Wigand and Copado Mejías, Jesús}, booktitle = {18. Internationale ILIAS-Konferenz}, date =
{2019-09-26/2019-09-27}, faupublication = {yes}, peerreviewed = {unknown}, title = {{Sage} als {Content}-{Element} in {ILIAS} nutzen}, venue = {Dresden}, year = {2019} } @incollection
{faucris.122402544, address = {Wien}, author = {Jahn, Johannes}, booktitle = {Mathematics of Multi Objective Optimization}, editor = {P. Serafini}, faupublication = {no}, note =
{UnivIS-Import:2015-04-17:Pub.1985.nat.dma.pama21.scalar}, pages = {45-88}, peerreviewed = {unknown}, publisher = {Springer}, title = {{Scalarization} in {Multi} {Objective} {Optimization}}, year =
{1985} } @article{faucris.110288904, abstract = {In this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem.
This will be done for weakly minimal and minimal solutions, respectively. Finally the results will be applied to a certain class of approximation problems. © 1984 The Mathematical Programming
Society, Inc.}, author = {Jahn, Johannes}, doi = {10.1007/BF02592221}, faupublication = {no}, journal = {Mathematical Programming}, keywords = {Approximation; Multi-Objective Programming; Vector
Optimization}, note = {UnivIS-Import:2015-03-05:Pub.1984.nat.dma.pama21.scalar}, pages = {203-218}, peerreviewed = {Yes}, title = {{Scalarization} in {Vector} {Optimization}}, volume = {29}, year =
{1984} } @article{faucris.116397864, abstract = {In this paper, we propose second-order epiderivatives for set-valued maps. By using these concepts, second-order necessary optimality conditions and a
sufficient optimality condition are given in set optimization. These conditions extend some known results in optimization. © 2005 Springer Science+Business Media, Inc.}, author = {Jahn, Johannes and
Khan, Akhtar Ali and Zeilinger, P.}, doi = {10.1007/s10957-004-1841-0}, faupublication = {yes}, journal = {Journal of Optimization Theory and Applications}, keywords = {Optimality conditions;
Second-order epiderivatives; Set optimization}, note = {UnivIS-Import:2015-03-09:Pub.2005.nat.dma.pama21.second}, pages = {331-347}, peerreviewed = {Yes}, title = {{Second}-{Order} {Optimality}
{Conditions} in {Set} {Optimization}}, volume = {125}, year = {2005} } @phdthesis{faucris.108545404, author = {Jahn, Johannes}, faupublication = {no}, peerreviewed = {automatic}, school =
{Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{Sequentieller} {Innerer}-{Punkt}-{Algorithmus} zur {Lösung} nichtlinearer {Optimierungsprobleme}}, year = {1978} } @article
{faucris.111888304, abstract = {In this paper set-semidefinite optimization is introduced as a new field of vector optimization in infinite dimensions covering semidefinite and copositive
programming. This unified approach is based on a special ordering cone, the so-called K-semidefinite cone for which properties are given in detail. Optimality conditions in the KKT form and duality
results including the linear case are presented for K-semidefinite optimization problems. A penalty approach is developed for the treatment of the special constraint arising in K-semidefinite
optimization problems. © Heldermann Verlag.}, author = {Eichfelder, Gabriele and Jahn, Johannes}, faupublication = {yes}, journal = {Journal of Convex Analysis}, keywords = {Convex analysis;
Copositive programming; Semidefinite programming; Vector optimization}, note = {UnivIS-Import:2015-03-09:Pub.2008.nat.dma.pama21.setsem}, pages = {767-801}, peerreviewed = {Yes}, title = {{Set}-
{Semidefinite} {Optimization}}, volume = {15}, year = {2008} } @incollection{faucris.122944404, address = {Dordrecht}, author = {Jahn, Johannes}, booktitle = {Encyclopedia of Optimization}, editor =
{C.A. Floudas, P.M. Pardalos}, faupublication = {yes}, note = {UnivIS-Import:2015-04-20:Pub.2001.nat.dma.pama21.setval}, pages = {3486 - 3488}, peerreviewed = {unknown}, publisher = {Kluwer}, title =
{{Set}-{Valued} {Optimization}}, year = {2009} } @incollection{faucris.111077384, abstract = {In this review article the theoretical foundations for shape-topological sensitivity analysis of elastic
energy functional in bodies with nonlinear cracks and inclusions are presented. The results obtained can be used to determine the location and the shape of inclusions which influence in a desirable
way the energy release at the crack tip. In contrast to the linear theory, where in principle, crack lips may mutually penetrate, here we employ nonlinear elliptic boundary value problems in
non-smooth domains with cracks with non-penetration contact conditions across the crack lips or faces. A shape-topological sensitivity analysis of the associated variational inequalities is performed
for the elastic energy functional. Topological derivatives of integral shape functionals for variational inequalities with unilateral boundary conditions are derived. The closed form results are
obtained for the Laplacian and linear elasticity in two and three spatial dimensions. Singular geometrical perturbations in the form of cavities or inclusions are considered. In the variational
context the singular perturbations are replaced by regular perturbations of bilinear forms. The obtained expressions for topological derivatives are useful in numerical method of shape optimization
for contact problems as well as in passive control of crack propagation.}, author = {Leugering, Günter and Sokolowski, Jan and Zochowski, Antoni}, booktitle = {Shape- and Topology Optimization for
Passive Control of Crack Propagation - New Trends in Shape Optimization}, doi = {10.1007/978-3-319-17563-8{\_}7}, editor = {Aldo Pratelli; Günter Leugering}, faupublication = {yes}, isbn =
{978-3-319-17563-8}, keywords = {Frictionless contact;Elastic bodies with cracks;Signorini conditions on cracks;Variational inequality;Shape functional;Shape sensitivity;Topological sensitivity;
Domain decomposition;Steklov-Poincare operator;Contact problems}, month = {Jan}, pages = {141-197}, peerreviewed = {unknown}, publisher = {Springer International Publishing}, series = {International
Series of Numerical Mathematics}, title = {{Shape}- and {Topology} {Optimization} for {Passive} {Control} of {Crack} {Propagation}}, url = {http://link.springer.com/chapter/10.1007/978-3-319-17563-8
{\_}7}, volume = {166}, year = {2015} } @article{faucris.107383364, abstract = {The aim of our work is to develop optimal dielectric composite structures with specific qualities. The task is to
design interfaces of given material components such that the originated structure attains certain optical properties. Propagation of the electromagnetic waves in the composite is described by the
Helmholtz equation. Success of the structure is enumerated by the objective function which is to be minimized. Interfaces of the given materials are parametrized by the cubic B–spline curves. The
design variables are afterwards the positions of B–spline control points. For objective function evaluation one forward computation of the Helmholtz equation is needed. To get the sensitivity of the
objective function we solve the backward (adjoint) equation.}, author = {Seifrt, Franti ek and Leugering, Günter and Rohan, Eduard}, doi = {10.1002/pamm.200810705}, faupublication = {yes}, journal =
{Proceedings in Applied Mathematics and Mechanics}, pages = {10705-10706}, peerreviewed = {Yes}, title = {{Shape} optimization for the {Helmholtz} equation}, url = {http://onlinelibrary.wiley.com/doi
/10.1002/pamm.200810705/full}, volume = {8}, year = {2008} } @incollection{faucris.117650104, abstract = {We consider shape optimization for objects illuminated by light. More precisely, we focus on
time-harmonic solutions of the Maxwell system in curl-curl-form scattered by an arbitrary shaped rigid object. Given a class of cost functionals, including the scattered energy and the extinction
cross section, we develop an adjoint-based shape optimization scheme which is then applied to two key applications.}, author = {Semmler, Johannes and Pflug, Lukas and Stingl, Michael and Leugering,
Günter}, booktitle = {New Trends in Shape Optimization}, doi = {10.1007/978-3-319-17563-8{\_}11}, editor = {Aldo Pratelli, Günter Leugering}, faupublication = {yes}, keywords = {Partial Differential
Equations; Systems Theory; Control}, pages = {251-269}, peerreviewed = {unknown}, publisher = {Springer}, series = {International Series of Numerical Mathematics}, title = {{Shape} {Optimization} in
{Electromagnetic} {Applications}}, volume = {166}, year = {2015} } @article{faucris.107373904, abstract = {The optimization of shape and topology of piezo-patches or layered piezo-electrical material
attached to structural parts, such as elastic bodies, plates and shells, plays a major role in the design of smart structures, as piezo-mechanic-acoustic devices in loudspeakers or energy harvesters.
While the design for time-harmonic motions is genuinely frequency-dependent, as has been reported in the literature in the context of density optimization with the SIMP-method, time-varying
piezoelectric material has not been investigated with respect to the optimal design so far. Therefore, shape sensitivities for layered piezoelectric material and time-varying loads and charges are
derived in this paper. In particular, we provide the shape-derivatives for nested piezo-layers associated with a class of shape functional. More general layers can be dealt with similar approach.},
author = {Leugering, Günter and Novotny, A. A. and Perla Menzala, G. and Sokolowski, J.}, doi = {10.1002/mma.1324}, faupublication = {yes}, journal = {Mathematical Methods in the Applied Sciences},
pages = {2118--2131}, peerreviewed = {Yes}, title = {{Shape} sensitivity analysis of a quasi-electrostatic piezoelectric system in multilayered media}, volume = {33}, year = {2010} } @book
{faucris.109706124, abstract = {A review of results on first order shape-topological differentiability of energy functionals for a class of variational inequalities of elliptic type is presented. The
velocity method in shape sensitivity analysis for solutions of elliptic unilateral problems is established in the monograph (Sokołowski and Zolésio, Introduction to Shape Optimization: Shape
Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992). The shape and material derivatives of solutions to frictionless contact problems in solid mechanics are obtained. In this way the
shape gradients of the associated integral functionals are derived within the framework of nonsmooth analysis. In the case of the energy type functionals classical differentiability results can be
obtained, because the shape differentiability of solutions is not required to obtain the shape gradient of the shape functional (Sokołowski and Zolésio, Introduction to Shape Optimization: Shape
Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992). Therefore, for cracks the strong continuity of solutions with respect to boundary variations is sufficient in order to obtain first
order shape differentiability of the associated energy functional. This simple observation which is used in Sokołowski and Zolésio (Introduction to Shape Optimization: Shape Sensitivity Analysis,
Springer, Berlin/Heidelberg/New York, 1992) for the shape differentiability of multiple eigenvalues is further applied in Khludnev and Sokołowski (Eur. J. Appl. Math. 10:379–394, 1999; Eur. J. Mech.
A Solids 19:105–120, 2000) to derive the first order shape gradient of the energy functional with respect to perturbations of the crack tip. A domain decomposition technique in shape-topology
sensitivity analysis for problems with unilateral constraints on the crack faces (lips) is presented for the shape functionals.We introduce the Griffith shape functional as the distributed shape
derivative of the elastic energy evaluated in a domain with a crack, with respect to the crack length. We are interested in the dependence of this functional on domain perturbations far from the
crack. As a result, the directional shape and topological derivatives of the nonsmooth Griffith shape functional are obtained with respect to boundary variations of an inclusion.}, author =
{Leugering, Günter and Sokolowski, Jan and Zochowski, Antoni}, doi = {10.1007/978-3-319-08025-3{\_}8}, faupublication = {yes}, keywords = {Conical differential of metric projection; Dirichlet Sobolev
space; Griffith criterium for crack propagation; Hadamard shape differentiability; Nonsmooth analysis; Shape gradient; Shape Hessian; Signorini variational inequality}, pages = {243-284},
peerreviewed = {Yes}, publisher = {Springer Verlag}, title = {{Shape}-topological differentiability of energy functionals for unilateral problems in domains with cracks and applications}, volume =
{101}, year = {2015} } @inproceedings{faucris.124195764, abstract = {We introduce the Griffith shape functional as the distributed shape derivative of the elastic energy evaluated in a domain with a
crack, with respect to the crack length. We are interested in the dependence of this functional on domain perturbations far from the crack. As a result, the directional shape and topological
derivatives of the nonsmooth Griffith shape functional are obtained with respect to the boundary variations of an inclusion. © 2013 West Pomeranian University of Technology.}, author = {Leugering,
Günter and Sokolowski, Jan and Zochowski, Antoni and Sokolowski, Jan and Zochowski, Antoni}, booktitle = {2013 18th International Conference on Methods and Models in Automation and Robotics, MMAR
2013}, faupublication = {yes}, isbn = {9781467355063}, keywords = {energy functional;crack with nonpenetration condition;shape gradient;topological derivative;Griffith's functional}, month = {Jan},
pages = {524-531}, peerreviewed = {Yes}, publisher = {IEEE SERVICE CENTER, 445 HOES LANE, PO BOX 1331, PISCATAWAY, NJ 08855-1331 USA}, title = {{Shape}-topological differentiability of energy
functionals in domains with cracks}, url = {https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=84893506756&origin=inward}, venue = {Miedzyzdroje}, year = {2013} } @article
{faucris.122687664, abstract = {We consider abstract optimal control problems in Banach spaces depending on a small parameter. Problems of this type appear in sensitivity analysis, discretization,
perturbation theory and homogenization of heterogenous material. We study the problem of passing to the limit within the framework of variational S-convergence. We derive conditions under which the
limiting problems can be made explicit, and we verify such conditions for a number of examples.}, author = {Kogut, Peter I. and Leugering, Günter}, doi = {10.1002/1522-2616(200201)233:1
<141::AID-MANA141>3.0.CO;2-I}, faupublication = {no}, journal = {Mathematische Nachrichten}, keywords = {homogenization; S-convergence; optimal control}, month = {Jan}, pages = {141-169},
peerreviewed = {Yes}, title = {{S}-homogenization of optimal control problems in {Banach} spaces}, url = {http://onlinelibrary.wiley.com/doi/10.1002/1522-2616(200201)
233:1%3C141::AID-MANA141%3E3.0.CO;2-I/abstract}, volume = {233 - 234}, year = {2002} } @article{faucris.214897680, abstract = {This article is concerned with the efficient and accurate simulation and
optimization of linear Timoshenko beam networks subjected to external loads. A solution scheme based on analytic ansatz-functions known to provide analytic solutions for the deformation and rotation
of a single beam with given boundary data is extended to the full network. It is demonstrated that the analytic approach is equivalent to a finite element (FE) method where only one element with a
suitably chosen shape function per beam is required. The solution of the FE-type system provides analytic solutions at the nodes, from which the solutions along the beams can be reconstructed.
Consequently analytic solutions for the network can be computed by a numerical scheme without applying a spacial discretization. While the assembly of the local stiffness matrices is slightly more
expensive compared to an FE model using, e.g., linear ansatz-functions, the complexity of the solution of the FE-system is not. This is particularly interesting for topology and material optimization
problems formulated on the network. In order to demonstrate the efficiency of the approach a numerical comparison to the case of linear ansatz-functions is provided followed by a series of studies
with topology and multi-material optimization problems on networks.}, author = {Kufner, Tobias and Leugering, Günter and Semmler, Johannes and Stingl, Michael and Strohmeyer, Christoph}, doi =
{10.1051/m2an/2018065}, faupublication = {yes}, journal = {Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique}, note = {CRIS-Team WoS
Importer:2019-03-29}, pages = {2409-2431}, peerreviewed = {Yes}, title = {{SIMULATION} {AND} {STRUCTURAL} {OPTIMIZATION} {OF} {3D} {TIMOSHENKO} {BEAM} {NETWORKS} {BASED} {ON} {FULLY} {ANALYTIC}
{NETWORK} {SOLUTIONS}}, volume = {52}, year = {2019} } @article{faucris.112318844, abstract = {In brittle composite materials, failure mechanisms like debonding of the matrix-fiber interface or fiber
breakage can result in crack deflection and hence in the improvement of the damage tolerance. More generally it is known that high values of fracture energy dissipation lead to toughening of the
material.Ouraimis toinvestigate the influence of material parameters and geometrical aspects of fibers on the fracture energy as well as the crack growth for given load scenarios. Concerning
simulations of crack growth the cohesive element method in combination with the Discontinuous Galerkin method provides a framework to model the fracture considering strength, stiffness and failure
energy in an integrated manner. Cohesive parameters are directly determined by DFT supercell calculations.We perform studies with prescribed crack paths as well as free crack path simulations. In
both cases computational results reveal that fracture energy depends on both the material parameters but also the geometry of the fibers. In particular it is shown that the dissipated energy can be
increased by appropriate choices of cohesive parametersof the interface and geometrical aspects of the fiber. In conclusion, our results can help to guide themanufacturing process ofmaterials with a
high fracture toughness.}, author = {Prechtel, Marina and Ronda, Pavel Leiva and Janisch, Rebecca and Hartmaier, Alexander and Leugering, Günter and Steinmann, Paul and Stingl, Michael}, doi =
{10.1007/s10704-010-9552-z}, faupublication = {yes}, journal = {International Journal of Fracture}, note = {UnivIS-Import:2015-03-09:Pub.2010.tech.FT.FT-TM.simula}, peerreviewed = {Yes}, title =
{{Simulation} of fracture in heterogeneous elastic materials with cohesive zone models}, year = {2010} } @article{faucris.264063015, abstract = {Singularly perturbed reaction–diffusion equations on a
star graph (having k + 1 nodes and k edges) resulting in a system with k individual partial differential equations along the edges with coupling conditions at the common junction are presented. In
the singular limit, as the diffusion parameter tends to zero, possibly individually along each edge, boundary layers may occur at the multiple nodes as well as at the simple nodes. Numerically, the
proposed equations are solved using central finite difference schemes on properly extended Shishkin meshes. Error estimates are discussed and validated by solving a test problem on a graph with three
edges (tripod). A more general graph problem with eight edges and three connecting nodes has also been solved numerically.}, author = {Kumar, Vivek and Leugering, Günter}, doi = {10.1002/mma.7749},
faupublication = {yes}, journal = {Mathematical Methods in the Applied Sciences}, keywords = {central finite difference schemes; k−star graph; networks; Shishkin meshes; tree-like graphs}, note =
{CRIS-Team Scopus Importer:2021-09-17}, peerreviewed = {Yes}, title = {{Singularly} perturbed reaction–diffusion problems on a k-star graph}, year = {2021} } @article{faucris.222492419, author =
{Meßner, Arthur and Engel, Ulf and Kals, Roland and Vollertsen, Frank}, faupublication = {yes}, journal = {Journal of Materials Processing Technology}, note = {LFT Import::2019-07-16 (1974)}, pages =
{371-376}, peerreviewed = {Yes}, title = {{Size} {Effect} in the {FE} {Simulation} of {Micro} {Forming} {Processes}}, volume = {45}, year = {1994} } @article{faucris.121559504, abstract = {We
investigate a class of constrained inverse homogenization problems. The complexity of the topological solution is restricted using slope constraint regularization. We show existence of the solution
for the inverse optimization problem in function space and outline a converging approximation scheme. We demonstrate how a proper numerical implementation can lead to a stable material design
approach. We finally describe results for a comprehensive set of numerical test cases.}, author = {Schury, Fabian and Stingl, Michael and Wein, Fabian}, doi = {10.1007/s00158-012-0795-3},
faupublication = {yes}, journal = {Structural and Multidisciplinary Optimization}, keywords = {Slope constraints;Inverse homogenization;Topology optimization;SIMP}, pages = {813-827}, peerreviewed =
{Yes}, title = {{Slope} constrained material design}, volume = {46}, year = {2012} } @article{faucris.115874264, abstract = {For p is not 2, only few results abaout analytic solutions of problems of
optimal control of distributed parameter systems with LP-norm have been reported in the literature. In this paper we consider such a problem for the wave equation, where the derivative of the state
is controlled at both boundaries. The aim is to steer the system from a position of rest to a constant terminal state in a given finite time. Also more general final configurations are considered.
The objective function that is to be minimized is the maximum of the L-p-norms of the control functions at both boundaries. It is shown that the analytic solution is, in fact, independent of the
choice of the p norm that is minimized. So the optimal controls solve a problem of multicriteria optimization, with the L-p-norms as objective functions.}, author = {Gugat, Martin and Leugering,
Günter}, faupublication = {yes}, journal = {Computational and Applied Mathematics}, keywords = {optimal control;wave equation;analytic solution;distributed parameter systems;boundary control;robust
control; 90C31; 90C34; 49K40}, month = {Jan}, note = {UnivIS-Import:2015-03-09:Pub.2002.nat.dma.lama1.soluti}, pages = {227-244}, peerreviewed = {Yes}, title = {{Solutions} of {L}-p-norm-minimal
control problems for the wave equation}, volume = {21}, year = {2002} } @article{faucris.111069684, abstract = {This article presents some calculus rules for contingent epiderivatives of set-valued
maps. Among other results the main emphasis is focused on a formula for scalar multiplication, sum formulae and chain rules. The calculus of contingent cones and some inversion theorems are used as a
tool. Some applications are also given.}, author = {Jahn, Johannes and Khan, Akhtar Ali}, doi = {10.1080/0233193031000079865}, faupublication = {yes}, journal = {Optimization}, keywords = {Adjacent
cone; Chain rule; Clarke's tangent cone; Contingent cone; Contingent epiderivative; Derivability; Optimality conditions; Sum formula}, note =
{UnivIS-Import:2015-03-09:Pub.2003.nat.dma.pama21.someca}, pages = {113-125}, peerreviewed = {Yes}, title = {{Some} {Calculus} {Rules} for {Contingent} {Epiderivatives}}, volume = {52}, year = {2003}
} @article{faucris.115106024, abstract = {In this paper several kinds of optima of a vector optimization problem are investigated; these are minimal, weakly minimal, strongly minimal and properly
minimal solutions. For these optima various optimality conditions resulting from a scalarization approach are presented. Furthermore some numerical aspects are discussed. © 1985 Springer-Verlag.},
author = {Jahn, Johannes}, doi = {10.1007/BF01719756}, faupublication = {no}, journal = {Or Spectrum}, note = {UnivIS-Import:2015-03-05:Pub.1985.nat.dma.pama21.somech}, pages = {7-17}, peerreviewed =
{Yes}, title = {{Some} {Characterizations} of the {Optimal} {Solutions} of a {Vector} {Optimization} {Problem}}, volume = {7}, year = {1985} } @inproceedings{faucris.107051164, abstract = {The
problem of exact null controllability of normalized strings and beams subject to boundary controls in an L/sup p/(0,T)-space, where p is larger than or equal to two, is discussed. In particular, the
case of L/sup infinity /-controls is of great importance in the applications. It is shown that subtle questions about Fourier series come into play in this problem. In particular, dealing with beam
related control problems, lacunary Fourier series play a dominant role; this refers to the notions of (p,q)-sets. Various results on exact controllability with L/sup p/-controls of special string and
beam models are given.}, author = {Leugering, Günter}, booktitle = {Decision and Control, 1990., Proceedings of the 29th IEEE Conference}, date = {1990-12-05/1990-12-07}, doi = {10.1109/
CDC.1990.203793}, editor = {IEEE}, faupublication = {no}, month = {Jan}, peerreviewed = {unknown}, publisher = {IEEE Xplore}, title = {{SOME} {REMARKS} {ON} {EXACT} {CONTROLLABILITY} {OF} {STRINGS}
{AND} {BEAMS} {WITH} {BOUNDARY} {CONTROLS} {IN} {LP}({O},{T}), {P}-{GREATER}-{THAN}-{OR}-{EQUAL}-{TO}-2}, url = {http://ieeexplore.ieee.org/document/203793/}, year = {1990} } @misc{faucris.117917184,
author = {Elliot, C and Hintermüller, Michael and Leugering, Günter and Sokolowski, Jan}, doi = {10.1080/10556788.2011.614721}, faupublication = {yes}, month = {Jan}, peerreviewed = {automatic},
title = {{Special} issue on advances in shape and topology optimization: theory, numerics and new applications areas - {Foreword}}, url = {http://www.tandfonline.com/doi/full/10.1080/
10556788.2011.614721}, year = {2011} } @article{faucris.119291084, author = {Jahn, Johannes and Chen, G.Y.}, faupublication = {yes}, journal = {Mathematical Methods of Operations Research}, note =
{UnivIS-Import:2015-03-05:Pub.1998.nat.dma.pama21.specia}, pages = {151-285}, peerreviewed = {Yes}, title = {{Special} {Issue} on '{Set}-{Valued} {Optimization}'}, volume = {48}, year = {1998} }
@article{faucris.110327624, author = {Jahn, Johannes}, faupublication = {yes}, journal = {Mathematical Methods of Operations Research}, note =
{UnivIS-Import:2015-03-05:Pub.1991.nat.dma.pama21.specia}, pages = {175-265}, peerreviewed = {Yes}, title = {{Special} {Issue} on {Vector} {Optimization}}, volume = {35}, year = {1991} }
@inproceedings{faucris.119852304, author = {Hante, Falk and Amin, Saurabh and Bayen, Alexandre M.}, booktitle = {Decision and Control, 2008. CDC 2008, 47th IEEE Conference on}, faupublication =
{yes}, pages = {2081--2086}, peerreviewed = {unknown}, title = {{Stability} analysis of linear hyperbolic systems with switching parameters and boundary conditions}, year = {2008} } @inproceedings
{faucris.118588184, abstract = {We consider the subcritical gas flow through star-shaped pipe networks. The gas flow is modeled by the isothermal Euler equations with friction. We stabilize the
isothermal Euler equations locally around a given stationary state on a finite time interval. For the stabilization we apply boundary feedback controls with time-varying delays. The delays are given
by C ^1-functions with bounded derivatives. In order to analyze the system evolution, we introduce an L ^2-Lyapunov function with delay terms. The boundary controls guarantee the exponential decay of
the Lyapunov function with time. © 2013 IFIP International Federation for Information Processing.}, address = {Berlin, Heidelberg}, author = {Gugat, Martin and Dick, Markus and Leugering, Günter},
booktitle = {System Modeling and Optimization}, date = {2011-09-12/2011-09-16}, doi = {10.1007/978-3-642-36062-6{\_}26}, faupublication = {yes}, keywords = {boundary feedback stabilization; Euler
equations; gas network; Lyapunov function; star-shaped network; time-varying delay}, note = {UnivIS-Import:2015-04-16:Pub.2013.nat.dma.zentr.stabil}, pages = {255-265}, peerreviewed = {Yes},
publisher = {Springer Verlag}, series = {IFIP Advances in Information and Communication Technology}, title = {{Stabilization} of the {Gas} {Flow} in {Star}-{Shaped} {Networks} by {Feedback}
{Controls} with {Varying} {Delay}}, url = {http://link.springer.com/chapter/10.1007/978-3-642-36062-6{\_}26}, venue = {Berlin}, volume = {391}, year = {2013} } @inproceedings{faucris.118314284,
abstract = {In the application of feedback controls, the computation of the controls may cause a delay. For vibrating systems, a constant delay can destroy the stabilizing effect of the control. To
avoid this problem we consider a feedback where a certain delay is a part of the control law and not a perturbation. We consider a string that is fixed at one end and controlled with a boundary
feedback with constant delay at the other end. We show the exponential stability of this system that is governed by the wave equation. Moreover, we show the robustness of the stability with respect
to variations in time of the feedback parameter that appears as a factor in the control law. ©2010 IEEE.}, author = {Gugat, Martin}, booktitle = {15th International Conference on Methods and Models
in Automation and Robotics (MMAR), 2010}, date = {2010-08-23/2010-08-26}, doi = {10.1109/MMAR.2010.5587248}, editor = {IEEE}, faupublication = {yes}, isbn = {978-1-4244-7828-6}, keywords =
{asymptotic stability, delays, feedback, vibration control, vibrations, wave equation}, note = {UnivIS-Import:2015-04-16:Pub.2010.nat.dma.zentr.stabil}, pages = {144-147}, peerreviewed = {unknown},
title = {{Stabilizing} a {Vibrating} {String} by {Time} {Delay}}, url = {http://ieeexplore.ieee.org/search/srchabstract.jsp?tp=&arnumber=5587248&
queryText%3DGugat%26openedRefinements%3D*%26searchField%3DSearch+All}, venue = {Miedzyzdroje}, year = {2010} } @article{faucris.112245584, abstract = {We consider a star-shaped network consisting of
a single node with N ≥ 3 connected arcs. The dynamics on each arc is governed by the wave equation. The arcs are coupled at the node and each arc is controlled at the other end. Without assumptions
on the lengths of the arcs, we show that if the feedback control is active at all exterior ends, the system velocity vanishes in finite time. In order to achieve exponential decay to zero of the
system velocity, it is not necessary that the system is controlled at all N exterior ends, but stabilization is still possible if, from time to time, one of the feedback controllers breaks down. We
give sufficient conditions that guarantee that such a switching feedback stabilization where not all controls are necessarily active at each time is successful. © American Institute of Mathematical
Sciences.}, author = {Gugat, Martin and Sigalotti, Mario}, doi = {10.3934/nhm.2010.5.299}, faupublication = {yes}, journal = {Networks and Heterogeneous Media}, keywords = {Feedback stabilization of
pdes; Hyperbolic pde; Network; Robustness; Switching control; Switching feedback; Wave equation}, note = {UnivIS-Import:2015-03-09:Pub.2010.nat.dma.zentr.starso}, pages = {299-314}, peerreviewed =
{Yes}, title = {{Stars} of vibrating strings: {Switching} boundary feedback stabilization}, url = {http://www.aimsciences.org/journals/displayArticles.jsp?paperID=5215}, volume = {5}, year = {2010} }
@article{faucris.204141165, abstract = {We introduce a stationary model for gas flow based on simplified isothermal Euler equations in a non-cycled pipeline network. Especially the problem of the
feasibility of a random load vector is analyzed. Feasibility in this context means the existence of a flow vector meeting these loads, which satisfies the physical conservation laws with box
constraints for the pressure. An important aspect of the model is the support of compressor stations, which counteract the pressure loss caused by friction in the pipes. The network is assumed to
have only one influx node; all other nodes are efflux nodes. With these assumptions the set of feasible loads can be characterized analytically. In addition we show the existence of optimal solutions
for some optimization problems with probabilistic constraints. A numerical example based on real data completes this paper.
shWwWe present the analysis for finding optimal locations and rotations of anisotropic material inclusions in a matrix material by using the polarization matrix. We compare different types of cost
functionals, in particular local ones, and show their respective differences. We use the Eshelby theorem and the representation of stresses based on the link matrix. As an analytical model reduction
technique, this allows for efficent numerical computation which is demonstrated for two selected examples.
}, author = {Leugering, Günter and Nazarov, Sergei and Schury, Fabian and Stingl, Michael}, doi = {10.1137/110823110}, faupublication = {yes}, journal = {SIAM Journal on Applied Mathematics},
keywords = {asymptotic analysis; Eshelby theorem; optimization; elastic patch; pointwise error estimates; integral and local cost functionals}, pages = {512-534}, peerreviewed = {Yes}, title = {{The}
{Eshelby} {Theorem} and {Application} to the optimization of an {Elastic} {Patch}}, volume = {72}, year = {2012} } @article{faucris.107390184, abstract = {We prove the Eshelby theorem for an
ellipsoidal piezoelectric inclusion in an infinite piezoelectric material. Explicit formulas for the link and polarization matrices are derived. Passing to the limits with respect to parameters in
the corresponding equations, the result is extended to cases when either the inclusion or the surrounding material is purely elastic.}, author = {Leugering, Günter and Nazarov, S. A.}, doi = {10.1007
/s00205-014-0803-4}, faupublication = {yes}, journal = {Archive for Rational Mechanics and Analysis}, pages = {n/a}, peerreviewed = {Yes}, title = {{The} {Eshelby} {Theorem} and its {Variants} for
{Piezoelectric} {Media}}, year = {2014} } @article{faucris.120147544, abstract = {This short note deals with the issue of existence of contingent epiderivatives for set-valued maps defined from a
real normed space to the real line. A theorem of Jahn-Rauh [1], given for the existence of contingent epiderivatives, is used to obtain more general existence results. The strength and the
limitations of the main result are discussed by means of some examples.}, author = {Jahn, Johannes and Khan, Akhtar Ali}, doi = {10.1016/S0893-9659(03)90114-5}, faupublication = {yes}, journal =
{Applied Mathematics Letters}, keywords = {Contingent cone; Contingent epiderivative; Set-valued maps}, note = {UnivIS-Import:2015-03-09:Pub.2003.nat.dma.pama21.theexi}, pages = {1179-1185},
peerreviewed = {Yes}, title = {{The} {Existence} of {Contingent} {Epiderivatives} for {Set}-{Valued} {Maps}}, volume = {16}, year = {2003} } @incollection{faucris.118834364, address = {Berlin},
author = {Jahn, Johannes}, booktitle = {Multiple Criteria Decision Making, Theory and Application}, editor = {G. Fandel, T. Gal}, faupublication = {no}, note =
{UnivIS-Import:2015-04-17:Pub.1980.nat.dma.pama21.thehaa}, pages = {128-134}, peerreviewed = {unknown}, publisher = {Springer}, series = {Lecture Notes in Economics and Mathematical Systems}, title =
{{The} {Haar} {Condition} in {Vector} {Optimization}}, volume = {177}, year = {1980} } @article{faucris.109406264, abstract = {The one-dimensional isothermal Euler equations are a well-known model
for the flow of gas through a pipe. An essential part of the model is the source term that models the influence of gravity and friction on the flow. In general the solutions of hyperbolic balance
laws can blow-up in finite time. We show the existence of initial data with arbitrarily large C^1-norm of the logarithmic derivative where no blow up in finite time occurs. The proof is based upon
the explicit construction of product solutions. Often it is desirable to have such analytical solutions for a system described by partial differential equations, for example to validate numerical
algorithms, to improve the understanding of the system and to study the effect of simplifications of the model. We present solutions of different types: In the first type of solutions, both the flow
rate and the density are increasing functions of time. We also present a second type of solutions where on a certain time interval, both the flow rate and the pressure decrease. In pipeline networks,
the bi-directional use of the pipelines is sometimes desirable. In this paper we present a classical solution of the isothermal Euler equations where the direction of the gas flow changes. In the
solution, at the time before the direction of the flow is reversed, the gas flow rate is zero everywhere in the pipe.}, author = {Gugat, Martin and Ulbrich, Stefan}, doi = {10.1016/
j.jmaa.2017.04.064}, faupublication = {yes}, journal = {Journal of Mathematical Analysis and Applications}, keywords = {Global classical solutions; Ideal gas; Bi-directional flow; Transsonic flow},
pages = {439–452}, peerreviewed = {Yes}, title = {{The} isothermal {Euler} equations for ideal gas with source term: {Product} solutions, flow reversal and no blow up}, url = {http://
www.sciencedirect.com/science/article/pii/S0022247X17304274}, volume = {454}, year = {2017} } @article{faucris.110752884, abstract = {The known Lagrange multiplier rule is extended to set-valued
constrained optimization problems using the contingent epiderivative as differentiability notion. A necessary optimality condition for weak minimizers is derived which is also a sufficient condition
under generalized convexity assumptions.}, author = {Jahn, Johannes and Götz, A.}, faupublication = {yes}, journal = {SIAM Journal on Optimization}, keywords = {Convex and set-valued analysis;
Optimality conditions; Vector optimization}, note = {UnivIS-Import:2015-03-05:Pub.1999.nat.dma.pama21.thelag}, pages = {331-344}, peerreviewed = {Yes}, title = {{The} {Lagrange} {Multiplier} {Rule}
in {Set}-{Valued} {Optimization}}, volume = {10}, year = {1999} } @article{faucris.120853084, abstract = {In this paper, we present a model for the controlled flow of a fluid through a network of
canals using a coupled system of St. Venant equations. We then generalize in a variety of ways recent results of Coron, d'Andrea-Novel, and Bastin concerning the stabilizability around equilibrium of
the flow through a single channel. This work is based on the theory of quasilinear hyperbolic systems and, in particular, on a delicate result of Li Ta-tsien concerning the existence and decay of
global classical solutions.}, author = {Leugering, Günter and Schmidt, E. J. P. Georg}, doi = {10.1137/S0363012900375664}, faupublication = {no}, journal = {SIAM Journal on Control and Optimization},
keywords = {St. Venant equations;hyperbolic systems;stabilizability;control of canals; 35L45; 35L50; 35L65; 93C20}, pages = {164-180}, peerreviewed = {Yes}, title = {{The} modelling and stabilization
of flows in networks of open canals}, url = {http://epubs.siam.org/doi/abs/10.1137/S0363012900375664}, volume = {41}, year = {2002} } @article{faucris.110515064, abstract = {An algorithm for
constrained rational Chebyshev approximation is introduced that combines the idea of an algorithm due to Hettich and Zencke, for which superlinear convergence is guaranteed, with the auxiliary
problem used in the well-known original differential correction method. Superlinear convergence of the algorithm is proved. Numerical examples illustrate the fast convergence of the method and its
advantages compared with the algorithm of Hettich and Zencke. © J.C. Baltzer AG Science Publishers.}, author = {Gugat, Martin}, doi = {10.1007/BF02143129}, faupublication = {no}, journal = {Numerical
Algorithms}, keywords = {Newton's method; Parametric auxiliary problem; Parametric optimization; Rational Chebyshev approximation with constrained denominators; Superlinear convergence; 41A20; 65D15;
90C32; 90C34}, note = {UnivIS-Import:2015-03-05:Pub.1996.nat.dma.lama1.thenew}, pages = {107-122}, peerreviewed = {Yes}, title = {{The} {Newton} differential correction algorithm for rational
{Chebyshev} approximation with constrained denominators}, url = {http://link.springer.com/article/10.1007/BF02143129}, volume = {13}, year = {1996} } @incollection{faucris.118886284, address =
{Boston}, author = {Jahn, Johannes}, booktitle = {Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications}, editor = {T. Gal, T.J. Stewart, T. Hanne},
faupublication = {yes}, note = {UnivIS-Import:2015-04-20:Pub.1999.nat.dma.pama21.theory}, pages = {2-1 - 2-32}, peerreviewed = {unknown}, publisher = {Kluwer}, title = {{Theory} of {Vector}
{Maximization}: {Various} {Concepts} of {Efficient} {Solutions}}, year = {1999} } @article{faucris.121378444, abstract = {We present an algorithm for the solution of general inequality constrained
optimization problems. The algorithm is based upon an exact penalty function that is approximated by a family of smooth functions. We present convergence results. As numerical examples we treat state
constrained optimal control problems for elliptic partial differential equations. We compare the results with existing methods. © 2010 Taylor & Francis.}, author = {Gugat, Martin and Herty, Michael},
doi = {10.1080/10556780903002750}, faupublication = {yes}, journal = {Optimization Methods & Software}, keywords = {Exact penalty function; Optimal control problem; Optimization with partial
differential equations}, note = {UnivIS-Import:2015-03-09:Pub.2010.nat.dma.zentr.thesmo}, pages = {573-599}, peerreviewed = {Yes}, title = {{The} smoothed-penalty algorithm for state constrained
optimal control problems for partial differential equations}, url = {http://www.informaworld.com/smpp/content~db=all~content=a913030109?words=gugat}, volume = {25}, year = {2010} } @article
{faucris.230343947, abstract = {This article presents an extensive theoretical framework to mathematically defined and information-based routing operators, applied to the continuous-time dynamic
traffic assignment problem. Because of the difficulty of the mathematical framework required to provide existence and uniqueness proofs of the solution to the problem in the presence of a routing
operator at nodes, the approach is instantiated with a link model, consisting of a system of ordinary delay differential equations and modeling traffic flow macroscopically. The routing operators
distributing the incoming flow can encompass a wide range of information patterns, which can include past knowledge of the network state (statistical, or past deterministic information) up to real
time and thus satisfying a nonanticipative character. We show, for a rather broad class of routing operators, the existence and uniqueness of solutions on the full network. This framework can be
extended to more advanced traffic flow models such as partial differential equation models.
L[2] (O, T)-boundary controls. Applying Laplace-Transform-techniques it is shown that L[2] (O, l)-states are exactly controllable in finite time, depending on the speed of propagation of
singularities. Finally the existence of time-optimal controls respecting a given norm bound is shown.}, author = {Leugering, Günter}, doi = {10.1002/mma.1670090130}, faupublication = {no}, journal =
{Mathematical Methods in the Applied Sciences}, month = {Jan}, pages = {413-430}, peerreviewed = {Yes}, title = {{TIME} {OPTIMAL} {BOUNDARY} {CONTROLLABILITY} {OF} {A} {SIMPLE} {LINEAR}
{VISCOELASTIC} {LIQUID}}, url = {http://onlinelibrary.wiley.com/doi/10.1002/mma.1670090130/full}, volume = {9}, year = {1987} } @article{faucris.117917404, abstract = {It is shown that the vibrations
of a viscoelastic beam can be steered to rest in minimal time using a L[2](O,T)-boundary control realizing a prescribed norm-bound.}, author = {Leugering, Günter}, doi = {10.1007/BFb0043877},
faupublication = {no}, journal = {Lecture Notes in Control and Information Sciences}, month = {Jan}, pages = {535-541}, peerreviewed = {Yes}, title = {{TIME} {OPTIMAL} {BOUNDARY} {CONTROLLABILITY}
{OF} {A} {VISCOELASTIC} {BEAM}}, url = {http://link.springer.com/chapter/10.1007/BFb0043877}, volume = {84}, year = {1986} } @article{faucris.110779944, abstract = {The problem of time-optimal
control of linear hyperbolic systems is equivalent to the computation of the root of the optimal value function of a time-parametric program, whose feasible set is described by a countable system of
moment equations. To compute this root, discretized problems with a finite number of equality constraints can be used. In this paper, we show that on a certain time-interval, the optimal value
functions of the discretized problems converge uniformly to the optimal value function of the original problem. We also give sufficient conditions for Lipschitz and Hölder continuity of the optimal
value function of the original problem.}, author = {Gugat, Martin}, faupublication = {no}, journal = {Control and Cybernetics}, keywords = {Continuity; Discretization; Hölder condition; Lipschitz
condition; Moment problems; Optimal value function; Parametric optimization; Time-minimal control; Uniform convergence}, note = {UnivIS-Import:2015-03-05:Pub.1999.nat.dma.zentr.timepa}, pages =
{7-33}, peerreviewed = {Yes}, title = {{Time}-{Parametric} {Control}: {Uniform} {Convergence} of the optimal value functions of the discretized problems}, volume = {28}, year = {1999} } @article
{faucris.115130884, author = {Itou, H. and Leugering, Günter and Khludnev, A. M.}, doi = {10.1134/S1028335814090018}, faupublication = {yes}, journal = {Doklady Physics}, pages = {401-404},
peerreviewed = {Yes}, title = {{Timoshenko} {Thin} {Inclusions} in an {Elastic} {Body} with {Possible} {Delamination}}, volume = {59}, year = {2014} } @inproceedings{faucris.121306504, abstract =
{This paper aims at identifying appropriate bottom anti-reflective coatings (BARCs) for double patterning techniques such as Litho-Freeze-Litho-Etch (LFLE). A short introduction into the employed
optimization methodology, including variables, figures of merit, models and optimization algorithms is given. A study on the impact of a refractive index modulation caused by the first lithographic
step is presented. Several optimization surveys taking the index modulation into account are set forth, and the results are discussed. In addition to optimization procedures aiming at optimizing one
litho step at a time, a co-optimization study for both litho steps is proposed. Finally, two multi-objective optimization procedures that allow for a post-optimization exploration and selection of
optimum solutions are presented. Numerous solutions are discussed in terms of their anti-reflectance behavior and their manufacturing feasibility. © 2010 SPIE.}, author = {Jahn, Johannes and Erdmann,
Andreas and Fühner, Tim and Liu, S. and Shao, F. and Barenbaum, A.}, booktitle = {Proceedings of the SPIE}, date = {2010-02-23/2010-02-25}, doi = {10.1117/12.846441}, faupublication = {yes}, keywords
= {bottom anti-reflective coating (BARC); double patterning; lithography simulation; waferstack optimization}, note = {UnivIS-Import:2015-04-16:Pub.2010.nat.dma.pama21.topogr}, pages = {76403C},
peerreviewed = {unknown}, publisher = {SPIE}, title = {{Topography}-aware {BARC} optimization for double patterning}, venue = {San Jose}, volume = {7640}, year = {2010} } @article{faucris.123758624,
abstract = {We consider linear second order differential equations on metric graphs under given boundary and nodal conditions. We are interested in the problem of changing the topology of the
underlying graph in that we replace a multiple node by a subgraph or concentrate a subgraph to a single node. We wish to do so in an optimal fashion. More precisely, given a cost function we may look
for its sensitivity with respect to these operations in order to find an optimal topology of the graph. Thus, in essence, we are looking for the topological gradient for linear second order problems
on metric graphs.}, author = {Leugering, Günter and Sokolowski, J.}, doi = {10.1002/zamm.201000067}, faupublication = {yes}, journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik},
pages = {926--943}, peerreviewed = {Yes}, title = {{Topological} derivatives for networks of elastic strings}, volume = {91}, year = {2011} } @article{faucris.110144804, abstract = {We consider
elliptic problems on graphs under given loads and bilateral contact conditions. We ask the question: which graph is best suited to sustain the loads and the constraints. More precisely, given a cost
function we may look at a multiple node of the graph with edge degree q and ask as to whether that node should be resolved into a number of nodes of edge degree less than q, in order to decrease the
cost. With this question in mind, we are looking into the sensitivity analysis of a graph carrying a second order elliptic equation with respect to changing its topology by releasing nodes with high
edge degree or including an edge. With the machinery at hand developed here, we are in the position to define the topological gradient of an elliptic problem on a graph.}, author = {Leugering, Günter
and Sokolowski, Jan}, faupublication = {yes}, journal = {Control and Cybernetics}, keywords = {differential equations on metric graphs;obstacles;topology optimization;asymptotic analysis}, month =
{Jan}, pages = {971-997}, peerreviewed = {Yes}, title = {{Topological} sensitivity analysis for elliptic problems on graphs}, url = {https://hal.archives-ouvertes.fr/hal-00261861/}, volume = {37},
year = {2008} } @incollection{faucris.123275504, address = {Berlin Heidelberg}, author = {Leugering, Günter and Martin, Alexander and Stingl, Michael}, booktitle = {Produktionsfaktor Mathematik - Wie
Mathematik Technik und Wirtschaft Bewegt}, doi = {10.1007/978-3-540-89435-3{\_}14}, editor = {Martin Grötschel, Klaus Lucas, Volker Mehrmann}, faupublication = {yes}, pages = {323-338}, peerreviewed
= {unknown}, publisher = {Springer}, title = {{Topologie} und {Dynamische} {Netzwerke}: {Anwendungen} {Der} {Optimierung} {MIT} {Zukunft}}, year = {2009} } @incollection{faucris.120543104, author =
{Leugering, Günter and Martin, Alexander and Stingl, Michael}, booktitle = {Produktionsfaktor Mathematik}, editor = {M. Grötschel, K. Lucas, V. Mehrmann}, faupublication = {yes}, pages = {323 --
340}, peerreviewed = {Yes}, series = {acatech diskutiert}, title = {{Topologie} und dynamische {Netzwerke}: {Anwendungen} der {Zukunft}}, year = {2008} } @incollection{faucris.117664184, abstract =
{The optimal design and control of infrastructures, e.g. in traffic control, water-supply, sewer-systems and gas-pipelines, the optimization of structures, form and formation of materials, e.g. in
lightweight structures, play a predominant role in modern fundamental and applied research. However, until very recently, simulation-based optimization has been employed in the sense that parameters
are being adjusted in a forward simulation using either ‘trial-and-error’ or a few steps of a rudimentary unconstrained derivative-free and mostly stochastic optimization code. It has become clear by
now that instead often a model-based and more systematic constrained optimization that exploits the structure of the problem under consideration may outperform the former more naive approaches. Thus,
modern mathematical optimization methods respecting constraints in state and design variables can be seen as a catalyst for recent and future technologies. More and more success stories can be
detected in the literature and even in the public press which underline the role of optimization as a key future technology. In particular, optimization with partial differential equations (PDEs) as
constraints, or in other words ‘PDE-constrained optimization’ has become research topic of great influence. A DFG-Priority-Program (PP) has been established by the German Science Foundation (DFG) in
2006 in which well over 25 project are funded throughout Germany. The PP focuses on the interlocking of fundamental research in optimization, modern adaptive, hierarchical and structure exploiting
algorithms, as well as visualization and validation. With similar goals in mind, a European network within the European Science Foundation (ESF) ‘PDE-constrained Optimization’ has been recently
established that provides a European platform for this cutting edge technology. In this report the authors dwell on exemplary areas of their expertise within applications that are already important
and will increasingly dominate future developments in mechanical and civil engineering. These applications are concerned with optimal material and design in material sciences and light-weight
structures as well as real-time capable optimal control of flows in transportation systems such as gas-pipeline networks. ‘Advanced Materials’, ‘Energy-Efficiency’ and ‘Transport’ are key problems
for the future society which definitely deserve public funding by national and international agencies.}, address = {Berlin Heidelberg}, author = {Leugering, Günter and Martin, Alexander and Stingl,
Michael}, booktitle = {Production Factor Mathematics}, doi = {10.1007/978-3-642-11248-5{\_}14}, editor = {Martin Grötschel, Klaus Lucas, Volker Mehrmann}, faupublication = {yes}, pages = {263-276},
peerreviewed = {unknown}, publisher = {Springer}, title = {{Topology} and {Dynamic} {Networks}: {Optimization} with {Application} in {Future} {Technologies}}, year = {2010} } @article
{faucris.110373604, abstract = {Vibrational piezoelectric energy harvesters are devices which convert ambient vibrational energy into electric energy. Here we focus on the common cantilever type in
which an elastic beam is sandwiched between two piezoelectric plates. In order to maximize the electric power for a given sinusoidal vibrational excitation, we perform topology optimization of the
elastic beam and tip mass by means of the SIMP approach, leaving the piezoelectric plates solid. We are interested in the first and especially second resonance mode. Homogenizing the piezoelectric
strain distribution is a common indirect approach increasing the electric performance. The large design space of the topology optimization approach and the linear physical model also allows the
maximization of electric performance by maximizing peak bending, resulting in practically infeasible designs. To avoid such problems, we formulate dynamic piezoelectric stress constraints. The
obtained result is based on a mechanism which differs significantly from the common designs reported in literature.}, author = {Wein, Fabian and Kaltenbacher, Manfred and Stingl, Michael}, doi =
{10.1007/s00158-013-0889-6}, faupublication = {yes}, journal = {Structural and Multidisciplinary Optimization}, keywords = {Topology optimization;Energy harvester;Stress constraints}, pages =
{173-185}, peerreviewed = {Yes}, title = {{Topology} optimization of a cantilevered piezoelectric energy harvester using stress norm constraints}, volume = {48}, year = {2013} } @article
{faucris.117284244, abstract = {We present the topology optimization of an assembly consisting of a piezoelectric layer attached to a plate with support. The optimization domain is the piezoelectric
layer. Using the SIMP (Solid Isotropic Material with Penalization) method with forced vibrations by harmonic electrical excitation, we achieve a maximization of the dynamic displacement. We show that
the considered objective function can be used under certain boundary conditions to optimize the sound radiation. The vibrational patterns resulting from the optimization are analysed in comparison
with the modes from an eigenvalue analysis. Multiple-frequency optimization is achieved by adaptive weighted sums. As a second optimization criterion, a flat frequency response is integrated in the
optimization process. }, author = {Wein, Fabian and Kaltenbacher, Manfred and Leugering, Günter and Bänsch, Eberhard and Schury, Fabian}, doi = {10.3233/JAE-2009-1022}, faupublication = {yes},
journal = {International Journal of Applied Electromagnetics and Mechanics}, keywords = {Topology optimization; piezoelectric coupling; sound radiation}, month = {Jan}, pages = {201-221},
peerreviewed = {Yes}, title = {{Topology} optimization of a piezoelectric-mechanical actuator with single- and multiple-frequency excitation}, url = {http://content.iospress.com/articles/
international-journal-of-applied-electromagnetics-and-mechanics/jae01022}, volume = {30}, year = {2009} } @phdthesis{faucris.202429588, abstract = {Numerical topology optimization based on the ersatz
material model is very attractive in the research community and industry. Large scale nonlinear problems can be solved efficiently through the availability of appropriate optimizers, often resulting
in non-intuitive solutions. However, topology optimization has not yet been established in the design of practical sensors and actuators. To this end we perform a thorough analysis and discussion of
two exemplary piezoelectric devices, a single-frequency loudspeaker and a cantilevered energy harvester.
With respect to the loudspeaker a broad range of objective functions is compared and discussed, culminating in a fully coupled piezoelectric-mechanical-acoustic near field topology optimization
problem. Piezoelectric strain cancellation and acoustic short circuits need to be balanced with structural resonance in order to obtain close to resonance performance for almost arbitrary target
frequencies. Providing appropriate initial designs proved to be essential for robust optimization.
Cantilevered piezoelectric energy harvesters have been subject to various optimization approaches. However these have generally been based on reduced model assumptions. We present topology
optimization of a realistic cantilevered energy harvester model. It proved to be necessary to use advanced topology optimization techniques, stress constraints to enforce practically feasible designs
and Heaviside filtering for void features size control and for obtaining a black and white design pattern. To the best of our knowledge, this is the first time that dynamic piezoelectric stress
constraints have been formulated for topology optimization. The obtained result is mechanism-based and interpretable to manufacture. This appears to be a novel finding in the field of cantilevered
piezoelectric energy harvesting design.
Performing numerical experiments, we were surprised to observe pronounced piezoelectric self-penalization, which means optimal black and white solutions without penalizing design interpolation and
additional constraints beside box constraints on the design variable. This phenomenon is only rarely and briefly described in the literature. Within this thesis we perform initial heuristic steps in
the analysis of the self-penalization phenomenon, which indeed appears in many different topology optimization problems. Once self-penalization is rigorously understood, our vision is to find methods
supporting the self-penalizing effect and to obtain solutions potentially closer to the original problem than constrained and penalized ersatz problems. To this end we present oscillation
constraints, a feature size control with independent solid and void feature size without enforcing intermediate pseudo material.
An important point for me is furthermore, that the students will get a picture or a visual impression of mathematical subjects. What does the kernel or the range of a linear mapping mean? What is the
difference of an accumulation point an a limit? What does is mean in 2D?
In my presentation I like to present some examples for different mathematical topics like:
• How linear mapping acts on polygon
• Domain and range is not the same
• Sequences in 2d
• Temperature profile in a plate, cylinder and sphere
• Boxplot and dataset
JSXGraph is included in classical HTML-Pages as media content and through dedicated plug-ins for the LMS Moodle and ILIAS. During the “Mathematics for Engineers” (part 1-3) we provide additional
material for several topics, e.g. series, (un)-constrained optimization, integration, parametrization of curves and surfaces or differential equations. We see the advantage that the diagrams will
become an integral part of the learning modules and the learning unit does not look like patchwork. Further, the learners can focus on the content and do not have to bring additional software or to
login in a second learning platform. During presence lectures the lecturer can use the same LMS as the learners do for their follow-up work.
All examples will be published for further usage.
×2 hyperbolic balance laws. The proofs are based on Wave-Front Tracking and therefore we present detailed results on the Riemann problem first.}, address = {Basel}, author = {Gugat, Martin and Herty,
Michael and Klar, Axel and Leugering, Günter and Schleper, Veronika}, booktitle = {Constrained Optimization and Optimal Control for Partial Differential Equations}, doi = {10.1007/978-3-0348-0133-1{\
_}7}, faupublication = {yes}, isbn = {978-3-0348-0133-1}, keywords = {Hyperbolic conservation laws on networks; optimal control of networked systems; management of fluids in pipelines; 35L65; 49J20},
note = {UnivIS-Import:2015-04-20:Pub.2012.nat.dma.lama1.wellpo}, pages = {123-146}, peerreviewed = {unknown}, publisher = {Springer}, series = {ISNM}, title = {{Well}-posedness of networked
hyperbolic systems of balance laws}, url = {http://link.springer.com/chapter/10.1007/978-3-0348-0133-1{\_}7}, volume = {160}, year = {2011} } @article{faucris.284143069, abstract = {We study
distributed optimal control problems governed by time-fractional parabolic equations with time dependent coefficients on metric graphs, where the fractional derivative is considered in the Caputo
sense. Using the Galerkin method and compactness results, for the spatial part, and approximating the kernel of the time-fractional Caputo derivative by a sequence of more regular kernel functions,
we first prove the well-posedness of the system. We then turn to the existence and uniqueness of solutions to the distributed optimal control problem. By means of the Lagrange multiplier method, we
develop an adjoint calculus for the right Caputo derivative and derive the corresponding first order optimality system. Moreover, we propose a finite difference scheme to find the approximate
solution of the state equation and the resulting optimality system on metric graphs. Finally, examples are provided on two different graphs to illustrate the performance of the proposed difference
scheme.}, author = {Mehandiratta, Vaibhav and Mehra, Mani and Leugering, Günter}, doi = {10.1002/asjc.2958}, faupublication = {yes}, journal = {Asian Journal of Control}, note = {CRIS-Team WoS
Importer:2022-10-28}, peerreviewed = {Yes}, title = {{Well}-posedness, optimal control and discretization for time-fractional parabolic equations with time-dependent coefficients on metric graphs},
year = {2022} } @article{faucris.110283844, abstract = {In this paper the multiple regression analysis is considered. The regression coefficients obtained by this method are examined with respect to
their numerical stability. With simple stability considerations a criteria is given which can be used if an estimator is to evaluate. © 1981 Phydica-Verlag.}, author = {Jahn, Johannes}, doi =
{10.1007/BF01902874}, faupublication = {yes}, journal = {Metrika}, note = {UnivIS-Import:2015-03-05:Pub.1981.nat.dma.pama21.zurbew}, pages = {23-33}, peerreviewed = {Yes}, title = {{Zur} {Bewertung}
von {Schätzungen} bei der gewöhnlichen {Regressionsanalyse}}, volume = {28}, year = {1981} } @incollection{faucris.106876264, author = {Jahn, Johannes}, booktitle = {Operations Research Verfahren},
faupublication = {no}, note = {UnivIS-Import:2015-03-05:Pub.1979.nat.dma.pama21.zursta}, pages = {209-215}, peerreviewed = {unknown}, title = {{Zur} {Stabilität} von {Regressionskoeffizienten}},
volume = {33}, year = {1979} } @article{faucris.121409464, author = {Jahn, Johannes}, faupublication = {no}, journal = {Optimization}, note =
{UnivIS-Import:2015-03-05:Pub.1983.nat.dma.pama21.zurvek}, pages = {577-591}, peerreviewed = {Yes}, title = {{Zur} vektoriellen linearen {Tschebyscheff}-{Approximation}}, volume = {14}, year = {1983}
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A thermally insulated vessel contains 150 g of water class 11 physics JEE_Main
Hint: To find the required fraction of water turns into ice we assume m mass of water gets converted into ice and then we will use the principle of the calorimeter and we take the formula: \[{Q_
{req}} = m{L_v}\] and \[{Q_{rel}} = m{L_f}\], where \[{Q_{req}}\] and \[{Q_{rel}}\] are the heat required and heat release in the process respectively. m and m’ are the amount of mass of water to ice
and rest in the vessel respectively. \[{L_f}\] and \[{L_v}\] are the latent heat of fusion of ice and latent heat of vaporization of water.
Complete step by step answer:
Let ‘m’ mass of water get converted into ice.
To get the required mass of water to be converted into ice formula used: \[{Q_{req}} = m{L_v}\] and \[{Q_{rel}} = m'{L_f}\] here \[{Q_{req}}\] and \[{Q_{rel}}\] are the heat required and heat release
in the process respectively. m is the mass of water to be converted into ice. \[{L_f}\] and \[{L_v}\] are the latent heat of fusion of ice and latent heat of vaporization of water.
The total mass of water = 150 g
\[{L_v} = 2.0 \times {10^6}J{\text{ }}k{g^{ - 1}}\]
\[{L_f} = 3.36 \times {10^5}J{\text{ }}k{g^{ - 1}}\]
If m mass of water converted into ice then the rest amount of water in the container will be = (150 – m)g
Now heat required for evaporation-
\[{Q_{req}} = m{L_v}\]………… (i)
and heat released in the process-
\[{Q_{rel}} = \left( {150 - m} \right){L_f}\]………….(ii)
Applying the principle of calorimeter:
Heat gain = Heat loss
\[\left( {150 - m} \right){L_f} = m{L_v}\]…………. (iii)
\[ \Rightarrow m\left( {{L_v} + {L_f}} \right) = 150{L_f}\]
\[ \Rightarrow m = \dfrac{{150{L_f}}}{{\left( {{L_v} + {L_f}} \right)}}\]…………….(iv)
Substitute the given value of \[{L_v}\]and \[{L_f}\]eqn (iv), we get
\[ \Rightarrow m = \dfrac{{150 \times 3.36 \times {{10}^5}}}{{\left( {2.0 \times {{10}^6} + 3.36 \times {{10}^5}} \right)}}kg\]
\[ \Rightarrow m = 20{\text{ }}g\]
Hence, option (C) is the correct answer.
Note: In order to master this kind of problem one should have basic understanding of the principle of the calorimeter and its implementation on a given problem. Students often confuse the terms
latent heat of vaporization and latent heat of fusion. The latent heat of vaporization is the amount of heat required to convert the unit mass of a substance from liquid to vapor phase while the
latent heat of fusion is the amount of heat required to convert the unit mass of a substance from solid to a liquid phase. So it is recommended that students should have to practice a lot of
conceptual problems on this topic and build their fundamentals clearly.
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Median - Formula, Meaning, Example | How to Find Median? (2024)
Median represents the middle value for any group. It is the point at which half the data is more and half the data is less. Median helps to represent a large number of data points with a single data
point. The median is the easiest statistical measure to calculate. For calculation of median, the data has to be arranged in ascending order, and then the middlemost data point represents the median
of the data.
Further, the calculation of the median depends on the number of data points. For an odd number of data, the median is the middlemost data, and for an even number of data, the median is the average of
the two middle values. Let us learn more about median, calculation of median for even-odd number of data points, and median formula in the following sections.
1. What is Median?
2. Median Formula
3. How to Find Median?
4. FAQs on How to Find Median
What is Median?
Median is one of the three measures of central tendency. When describing a set of data, the central position of the data set is identified. This is known as the measure of central tendency. The three
most common measures of central tendency are mean, median, and mode.
Median Definition
The value of the middle-most observation obtained after arranging the data in ascending order is called the median of the data. Many an instance, it is difficult to consider the complete data for
representation, and here median is useful. Among the statistical summary metrics, the median is an easy metric to calculate. Median is also called the Place Average, as the data placed in the middle
of a sequence is taken as the median.
Median Example
Let's consider an example to figure out what is median for a given set of data.
• Step 1: Consider the data: 4, 4, 6, 3, and 2. Let's arrange this data in ascending order: 2, 3, 4, 4, 6.
• Step 2: Count the number of values. There are 5 values.
• Step 3: Look for the middle value. The middle value is the median. Thus, median = 4.
Median Formula
Using the median formula, the middle value of the arranged set of numbers can be calculated. For finding this measure of central tendency, it is necessary to write the components of the group in
increasing order. The median formula varies based on the number of observations and whether they are odd or even. The following set of formulas would help in finding the median of the given data.
Median Formula for Ungrouped Data
The following steps are helpful while applying the median formula for ungrouped data.
• Step 1: Arrange the data in ascending or descending order.
• Step 2: Secondly, count the total number of observations 'n'.
• Step 3: Check if the number of observations 'n' is even or odd.
Median Formula When n is Odd
The median formula of a given set of numbers, say having 'n' odd number of observations, can be expressed as:
Median = [(n + 1)/2]^th term
Median Formula When n is Even
The median formula of a given set of numbers say having 'n' even number of observations, can be expressed as:
Median = [(n/2)^th term + ((n/2) + 1)^th term]/2
Example: The age of the members of a weekend poker team has been listed below. Find the median of the above set.
{42, 40, 50, 60, 35, 58, 32}
Step 1: Arrange the data items in ascending order.
Original set: {42, 40, 50, 60, 35, 58, 32}
Ordered Set: {32, 35, 40, 42, 50, 58, 60}
Step 2: Count the number of observations. If the number of observations is odd, then we will use the following formula: Median = [(n + 1)/2]^th term
Step 3: Calculate the median using the formula.
Median = [(n + 1)/2]^th term
= (7 + 1)/2^th term = 4^th term = 42
Median = 42
Median Formula for Grouped Data
When the data is continuous and in the form of a frequency distribution, the median is calculated through the following sequence of steps.
• Step 1: Find the total number of observations(n).
• Step 2: Define the class size(h), and divide the data into different classes.
• Step 3: Calculate the cumulative frequency of each class.
• Step 4: Identify the class in which the median falls. (Median Class is the class where n/2 lies.)
• Step 5: Find the lower limit of the median class(l), and the cumulative frequency of the class preceding the median class (c).
Now, use the following formula to find the median value.
Application of Median Formula
Let us use the above steps in the following practical illustration to understand the application of the median formula.
Illustration: There are 5 top management employees in an organization. The salaries given to the employees are $5,000, $6,000, $4,000, $8,000, and $7,500. Using the median formula calculates the
median salary.
Solution: We will follow the given steps to find the median salary.
• Step 1: Sorting the given data in increasing order, $4,000, $5,000, $6,000, $7,500, and $8,000.
• Step 2: Total number of observations = 5
• Step 3: The given number of observations is odd.
• Step 4: Using median formula for odd observation, Median = [(n + 1)/2]^th term
• Median = [(5+1)/2]^th term. = 6/3 = 3^rd term. The third term is $6,000.
The median salary is $6,000.
How to Find Median?
We use a median formula to find the median value of given data. For a set of ungrouped data, we can follow the below-given steps to find the median value.
• Step 1: Sort the given data in increasing order.
• Step 2: Count the number of observations.
• Step 3: If the number of observations is odd use median formula: Median = [(n + 1)/2]^th term
• Step 4: If the number of observations is even use median formula: Median = [(n/2)^th term + (n/2 + 1)^th term]/2
Example: The height (in centimeters) of the members of a school football team have been listed below.
{142, 140, 130, 150, 160,135, 158,132}
Find the median of the above set.
Step 1:
Arrange the data items in ascending order.
Original set: {142, 140, 130, 150, 160, 135, 158,132}
Ordered Set: {130, 132, 135, 140, 142, 150, 158, 160}
Step 2:
Count the number of observations.
Number of observations, n = 8
If number of observations is even, then we will use the following formula:
Median = [(n/2)^th term + ((n/2) + 1)^th term]/2
Step 3:
Calculate the median using the formula.
Median = [(n/2)^th term + ((n/2) + 1)^th term]/2
Median = [(8/2)^th term + ((8/2) + 1)^th term]/2
= (4^th term + 5^th term)/2
= (140 + 142)/2
= 141
For a set of grouped data, we can follow the following steps to find the median:
When the data is continuous and in the form of a frequency distribution, the median is calculated through the following sequence of steps.
• Step 1: Find the total number of observations(n).
• Step 2: Define the class size(h), and divide the data into different classes.
• Step 3: Calculate the cumulative frequency of each class.
• Step 4: Identify the class in which the median falls. (Median Class is the class where n/2 lies.)
• Step 5: Find the lower limit of the median class(l), and the cumulative frequency(c).
• Step 6: Apply the formula for median for grouped data: Median \(= l + [\dfrac {\dfrac{n}{2}-c}{f}]\times h\)
We have seen examples to find median for ungrouped data in the previous section. Here's an example to calculate median for grouped data.
Example: Calculate the median for the following data:
Marks 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100
Number of students 5 20 35 7 3
We need to calculate the cumulative frequencies to find the median.
Marks Number of students Cumulative frequency
0 - 20 5 0 + 5 5
20 - 40 20 5 + 20 25
40 - 60 35 25 + 35 60
60 - 80 7 60 + 7 67
80 - 100 3 67 + 3 70
N = \(\sum f_i\) = 70
N/2 = 70/2 = 35
Median Class is 40 - 60
l = 40, f = 35, c = 25, h = 20 \(\)
Using Median formula:
Median \(= l + [\dfrac {\dfrac{n}{2}-c}{f}]\times h\)
= 40 + [(35 - 25)/35] × 20
= 40 + (10/35) × 20
= 40 + (40/7)
Median of Two Numbers
In an ordered series, the median is the number that is mid-way between the range extremes. It is not usually identical to the mean. Let's understand how to find the median. For a set of two values,
the median will be the same as the mean, or arithmetic average. For example, the numbers 2 and 10, both have a mean and a median of 6. Note that the median is the value at the mid-point of the
dataset, not the mid-point of the values. The mean is the arithmetic average: (10 + 2)/2 = 6. What if we add two more numbers, say 3 and 4? The median will be 3.5, but the mean will be (2 + 3 + 4 +
10)/4 = 4.75
Important Notes on Median:
The above content to find the median has been summarized in the form of the following points.
• Median is the central value of data (Positional Average).
• Data has to be arranged in ascending/descending order to find the middle value or median.
• Not every value is considered while calculating the median.
• Median doesn't get affected by extreme points.
Thinking Out of the Box:
Now it's time to apply the learned concepts of the median. Here's a question for you!
Question: Determine the median of the first five whole numbers. In a company, for each of the 10 employees working in a service upgrade process, the number of service upgrades sold are as follows:
34, 26, 30, 21, 25, 12, 18, 20, 19, 15. Find out the median number of service upgrades sold by the 10 employees?
Related Topics on Median:
• Mode
• Average
• Mean
• Geometric Mean
FAQs on Median
What Is Meant By Median in Statistics?
The value of the middle-most observation obtained after arranging the data in ascending or descending order is called the median of the data. When describing a set of data, the central position of
the data set is identified and used further in the median formula. This is known as the measure of central tendency. Median is an important measure of central tendency.
What is the Median Formula for Ungrouped Data?
The median formula for ungrouped data is totally dependent on the number of observations (n). If the number of observation is odd then the median formula is [Median = {(n + 1)/2} ^th term]. If the
number of observation is even then the median formula is [Median = ((n/2)^th term + (n/2 + 1)^th term)/2]
What is the Median Formula When Number of Observations is Even?
When the number of observation is even (n = even) then the median formula is [Median = ((n/2)^th term + (n/2 + 1)^thterm)/2]
What is the Median Formula When Number of Observations is Odd?
When the number of observation is odd (n = odd) then the median formula is [Median = {(n + 1)/2} ^th term]
What is the Difference Between Mean, Median, Mode, and Range?
The mean is the arithmetic average of a given dataset. The median is the middle score in a set of given numbers. The mode is the most frequently occurring value in a set of given numbers. The range
is the difference between the highest and the lowest values.
How to Find Mean, Mode, and Range?
The mode refers to the most repetitive number in the given dataset. The mean is the average of all numbers: Add all the values and divide the sum by the number of values. The range is the difference
between the highest and the lowest values.
How To Calculate Median?
The median of a dataset is calculated by following two simple steps. First, arrange the given data in ascending order. Next, we need to pick the middlemost data.
• For an even number of data points, there are two middle values, and we need to take the average of those two middle values.
• For the odd number of data points, there is only one middle data point and we can take it as the median of the data.
What is Mean vs Median?
The mean of the data is the average of the data and is equal to the sum of all the data values divided by the number of data points. The median of the data is the mid-value of the data, after
arranging the data in ascending order.
Is the Median Same as the Average?
The median of the data is different from the average. The median is the mid-value of the given data points, and the average is the value obtained by dividing the sum of the data values by the number
of data points. But for equally spaced numbers such as 2, 4, 6, 8, 10, the median and the average are the same, that is 6.
What are the Applications of Median?
Median is an important statistical measure that helps in representing a single value for a large number of data points. As an example, the data of height or the age of the students in a class is
represented by a single median value of the data.
How to Arrange Data in Ascending Order?
For arranging the data in ascending order we need to write the data starting with the smallest values and further include the data points with increasing order of their value.
Why Median is Called Positional Average?
The median falls in the middle when the data is arranged in an increasing or decreasing order. Hence the median is referred to as positional average. The median is the exact middle value, in case of
odd number of data points whereas for even number of data points, the median is the average of the two middle values.
What Does n Represent In Median Formula?
In the median formula 'n' represent the number of observations. When 'n' is odd number of observations then the median formula is [Median = {(n + 1)/2} ^th term]. When 'n' is is even number of
observations then the median formula is [Median = ((n/2)^th term + (n/2 + 1)^th term)/2].
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Computational Thinking: Definition, Characteristics - Sinaumedia
Computational Thinking: Definition, Characteristics
Computational Thinking – This globalization era will indeed continue to develop, including the technology will also continue to develop. Therefore, we will meet new technologies that are increasingly
sophisticated and increasingly we will feel that our lives and activities are becoming faster. This is because we must be able to keep up with the times and increasingly dynamic technology. If we
take too long to keep up with the times and technology, it is possible that we will be left behind.
We must be able to think quickly about what we should do in the future. In addition, must develop what we have done. For example, we write a letter, then we must be able to develop it into a word to
a sentence. If we can develop something, then we already have a way of thinking to move forward or be more dynamic.
This mindset is the same as how a technology works where it will accept tasks and complete them quickly. Things like this can happen because we are starting to live side by side with technology, and
like it or not and like it or not, we have to have a way of thinking that is almost the same as technology. This needs to be done so that we can keep abreast of the times and technological
It has become a common thing, for many people, that almost all of the daily activities are related to technology. In fact, some of the problems we are currently facing can sometimes be solved with
existing technology. Therefore, we should be able to apply ways of thinking like computer science (informatics) techniques. By applying this way of thinking, it will be easy for us to think
critically and creatively.
In this case, the technology in question is computer technology. The development of this computer always leads to a modern and faster direction, so that when we use it, the activities we carry out
will feel easier. In a life that we live, whether we use a computer or not, we must be able to think like a computer that is able to understand a thing or problem quickly, so that we can find a
solution to a problem quickly. This pattern of thinking is known as “computational thinking”.
Then, what is computational thinking according to experts and what are examples of computational thinking like? So, to get all the answers, you can see this review, Sinaumed’s. So, happy reading.
Definition of Computational Thinking
Because we live side by side with technology, we need to think like a machine that can move dynamically. Therefore, computational thinking can be a concept or a way to observe problems and find
solutions to these problems by applying computer science technology. By thinking computationally, someone will be able to observe problems, solve problems so that they can develop solutions from
solving problems.
Basically, computational thinking is indeed adapting a thought or way of working that comes from a computer. However, some people still think that computational thinking must use computer
applications. In fact what is meant by computational thinking is not having to use a computer.
The term Computational Thinking or shortened to CT or computational thinking was first introduced in general in 1980 and 1996 by Seymor Papert. Over time, in 2014, the British government began
bringing programming material into the curriculum from elementary to high school. The inclusion of programming material into the education curriculum so that students are familiar with technology
from an early age. In addition, students are also expected to be able to think computationally from an early age.
The program carried out by the British government was apparently supported by figures who have influence in the field of technology, such as Bill Gates, Mark Zuckerberg, and others. Facilities that
can support the process of learning activities are assisted by the company Google through online training so that teachers or educators can understand and master Computational Thinking (CT).
Basically, to think computationally is not easy or you could say it requires more effort. Even though it’s hard to do, we have to believe and believe that we can change our thinking patterns into
computational thinking patterns. Therefore, we must accustom ourselves to think computationally in any situation. If we are used to thinking computationally, then we will feel the positive impact,
namely being able to think quickly, easily, and precisely.
In order to get used to thinking computationally, one should be taught from an early age to think computationally. It would be nice if every school in Indonesia has started to incorporate a
programming curriculum into the elementary and middle school education curriculum, so that computational thinking patterns can be instilled from an early age.
Characteristics of Computational Thinking
After discussing the meaning of computational thinking, now the next discussion is computational thinking. The characteristics of computational thinking are as follows.
1. Fundamental Not Memorizing
The first characteristic of computational thinking is fundamental not memorizing. In this case, what is meant by fundamental is ability. Every human being who already has basic abilities means that
he can understand something well, so it will be easy to find a solution to a problem. In fact, that person can develop a solution from a problem solving
On the other hand, with someone who has abilities based on memorization, it is likely that it will be difficult to solve problems because it is possible to forget something that is understood.
Therefore, someone who already has basic abilities and already understands something without memorizing it, it can be said that this person can already think computationally.
2. According to the Concept of Not Programming
The second characteristic of computational thinking is that it corresponds to non-programming concepts. In other words, computers and science are not just computer programming, but we must be able to
think like people who are proficient in the world of computers and science. In fact, we should also understand the programs that are on the computer.
With this characteristic, one must be accustomed to using programs on computers from an early age, so that it will be easy to understand the concept of computational thinking. In addition, we will be
proficient in running computer programming faster. If you are proficient in using computer programming, it will be easy for you to keep up with the times and be able to adapt to technology.
3. Ideas and Not Things
The third characteristic of computational thinking is that it prioritizes ideas or ideas over things. In other words, in solving a problem at hand, it is better to prioritize using computational
concepts. Not only that, this idea or concept should also be used in daily activities, managing daily life, and used when carrying out social interactions with other people.
With this characteristic, you could say that this computational concept can be trained to get used to using it. This needs to be done because computational concepts can provide many benefits for the
lives we live. Not only that, computational concepts can develop our ability to understand a problem, so that we can find solutions to a problem easily.
4. Complementary
The fourth characteristic of computational thinking is the complementarity between engineering and mathematics. Complementary can be interpreted like computer science which is very closely related to
mathematical thinking. Not only complementing it, but we also have to get used to combining mathematical thinking with technical thinking.
When we complement and combine mathematical thinking and engineering thinking, we are indirectly able to distinguish various kinds of things that can benefit or harm us. In addition, we will easily
do something that is very related to mathematics, such as building a building that is done by an engineer or architect.
5. Must be able to operate a computer
The fifth characteristic of computational thinking is having to be able to operate a computer. As we know that computational thinking is adopted from computer science technology, so it is natural for
humans to be able to operate computers. Moreover, in this modern era, every individual should operate a computer.
If you can operate a computer, it will be easy for you to work in any field. In simple terms, proficiency in operating a computer, we have many choices to continue a career, such as in the fields of
law, health, education, business, to the arts.
6. Can be used by anyone and anywhere
The sixth characteristic of computational thinking is that it can be used by anyone and anywhere. In other words, computational thinking can appear by everyone, including yourself and computational
thinking can be used anywhere, such as schools, homes, offices, and so on. In fact, it’s even better to use the concept of computational thinking in every activity we do.
New computational thinking patterns can be well realized, if they meet real human efforts which then turn into something philosophical and explicit. In short, computational efforts or actions and
patterns of thinking must be well intertwined, so that a problem can be solved or solved properly too.
7. Applying the Way Humans Think Not the Way Computers Think
The seventh characteristic of computational thinking is applying the way humans think, not the way computers think. As previously explained, if computational thinking is a method or a person’s way of
solving or solving problems. Therefore, every human being must use his own way of thinking, not following the computer’s way of thinking.
With this characteristic, a person should begin to realize that he has greater abilities than a computer. Therefore, in solving a problem, humans must be aware that computers are controlled by
humans, not humans who are controlled by computers. By being aware of things like that, then a problem will be easy to solve or solve.
8. Is Challenging From an Intellectual Point of View
The eighth characteristic of computational thinking is that it is challenging from an intellectual point of view. In this characteristic, someone who thinks computationally will make every effort to
understand and solve a scientific problem. In other words, by thinking computationally, our curiosity and creativity will be honed properly.
If curiosity and creativity have developed, ideas and ideas for doing something or solving a problem will also develop. In addition, the insight that we have with the presence of curiosity, even we
are also able to think creatively, so that we will never run out of ideas or ideas.
Benefits of Computational Thinking
Computational thinking has several benefits, including:
1. Make it easy for us to solve large and complex problems in an effective and efficient way. In addition, complex problems can be solved properly, so that they become simple problems.
2. Train the brain to get used to start thinking mathematically, creatively, structured and logically.
3. Make it easier for someone to observe the problem and find a solution to the problem. The more solutions you have, the more effectively and efficiently a problem can be solved.
Computational Ways of Thinking
To make it easier to apply computational thinking in everyday life, we need to know the ways or stages for computational thinking. The following will explain computational thinking.
1. Decomposition
Decomposition is a method or concept that functions to find a solution from a complex and large problem into smaller problems. If a large and complex problem becomes small, then the problem is easy
to solve. In fact, decomposition can be used to make it easier for us to find and implement an innovation. For example, we sell a product, then for the product to be innovated, it is very likely that
the product will sell well.
2. Pattern Recognition
Pattern recognition is a method that utilizes a computer to find regularities in data and to obtain more important information in order to understand the regularities that have been found. This
pattern recognition is usually done when we recognize someone from their voice, face, even this pattern recognition can be used to predict the weather. In a natural phenomenon, pattern recognition
can actually be seen in the earth’s rotation pattern, constellation patterns, patterns on leaves, and so on.
3. Abstraction
Abstraction is a method of computational thinking that prioritizes things that are directly related to the problem at hand. In addition, this abstraction concept will leave various things that are
considered unable to be used to solve a problem.
4. Algorithm
An algorithm is a mindset that is used to plan systematic steps to solve a problem that is happening. Even though this algorithm is often associated with calculations, this thinking method can be
used to solve various kinds of problems that exist in everyday life.
In this increasingly modern and dynamic era, every human being should have a fast and dynamic mindset so as not to be left behind by other individuals. Therefore, every human being should be able to
think computationally, especially now that the use of technology is unavoidable in everyday life.
By thinking computationally, it will be easy for someone to observe problems, find solutions to problems, solve problems, and be able to develop solutions or solve problems. In addition,
computational thinking sharpens us to think more effectively and efficiently.
Source: From various sources
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Add Leading Zeros to a Sequence in Google Sheets
5 Ghani Rozaqi 01/01/2023 3:51:49 Add Leading Zeros to a Sequence in Google Sheets Add Leading Zeros to a Sequence in Google Sheets
Google Sheets
Add Leading Zeros to a Sequence in Google Sheets
In Google Sheets, it is possible to add leading zeros to a number or generate a sequence of numbers with leading zeros. Leading zeros are digits that are added to the beginning of a number to make it
a certain number of digits long. For example, if you want to add leading zeros to the number “1” so that it becomes four digits long, you can add three leading zeros, resulting in the number “0001”.
This can be useful if you want to maintain a consistent number of digits for a series of numbers or if you need to format numbers for a specific purpose. In this tutorial, we will show you how to add
leading zeros to a number or generate a sequence of numbers with leading zeros in Google Sheets.
1. Add Leading Zeros with the PAUSE DROP Function
First, you can use the apostrophe add leading zeros to a number. To do this, you can type ‘ (apostrophe) in a cell, where A1 is the cell containing the number you want to add leading zeros to and 4
is the total number of digits you want the resulting number to have (including the leading zeros). This function will add leading zeros to the number in A1 until it reaches the desired number of
2. Add Leading Zeros with the CONCATENATE Function
Alternatively, you can use the CONCATENATE function to add leading zeros to a number. To do this, you can type =CONCATENATE(“0000”, A1) in a cell, where A1 is the cell containing the number you want
to add leading zeros to. This function will add four leading zeros to the number in A1. You can change the number of leading zeros by adjusting the number of zeros in the string of characters.
3. Generate a Number Sequence with the SEQUENCE Function
To generate a sequence of numbers with leading zeros, you can use the SEQUENCE function. To do this, you can type =SEQUENCE(10, 1, 1, 1) in a cell, where 10 is the number of rows you want to
generate, 1 is the starting value, and 1 is the increment value. This function will generate a sequence of 10 numbers starting at 1 and increasing by 1 for each subsequent number.
4. Generate a Number Sequence with Leading Zeros and the SEQUENCE and CONCATENATE Functions
To combine the SEQUENCE function with the CONCATENATE function to generate a sequence of numbers with leading zeros, you can type =CONCATENATE(“0000”, SEQUENCE(10, 1, 1, 1)) in a cell. This will
generate a sequence of 10 numbers with leading zeros, starting at 0001 and increasing by 1 for each subsequent number. You can adjust the number of leading zeros by changing the number of zeros in
the string of characters.
5. Fill a Range of Cells with the Resulting Numbers
To fill a range of cells with the resulting numbers, you can select the cell with the formula and drag the fill handle (the small square in the bottom-right corner of the cell) to the desired range
of cells. The formula will be automatically adjusted for each cell in the range.
6. Adjust the Starting Value and Increment Value in the SEQUENCE Function
If you want to change the starting value or the increment value for the sequence, you can simply adjust the corresponding values in the SEQUENCE function. For example, to start the sequence at 10 and
increase by 2 for each subsequent number, you can use the formula =CONCATENATE(“0000”, SEQUENCE(10, 10, 2, 1)).
Categorised in: Google Sheets
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Is there an easy formula the calculate the floor area of a geodesic dome structure?
One of my fellow commercial real estate agents has a property for sale that contains a geodesic dome structure. Is there a relatively straightforward way to calculate the floor area?
Yes. Yes there is.
Area=3.14159 * radius^2.
Or, since it’s easier to measure the distance across the building, the diameter, you can use:
area = distance across[sup]2[/sup]*22/28
On a 40 ft. building you will be off by 1/2 ft. sq. from using the true value of π
That is, geodesic domes are approximations of spheres. The ideal dome uses flexible support members that form geodesics (“great circles”). Therefore, when constructed they form a sphere. The
intersection of a plane that is perpendicular to the sphere’s radius and bisects the sphere is a circle, so the floor area is the area of a circle (pi times radius squared).
However, this neglects the reality of geodesic domes in construction. Very few of them are “perfect” spheres. I have built model geodesics that use pentagons; others use alternating regular polygons
as components. So using the area of a circle is just an approximation.
By the way, a geodesic is a an arc on the circle formed when a plane intersects a sphere and contains the sphere’s center. Another way of looking at it is that a geodesic is the shortest distance
between any two points on a sphere.
The most common everyday occurrence of great circles is airplane routes, especially between continents.
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Handbook Of Philosophical Logic. Volume I: Elements Of Classical Logic [DJVU] [7kl504sndiu0]
E-Book Overview
The Handbook of Philosophical Logic is a unique systematic survey of the central areas of philosophical logic. Divided into four volumes, each devoted to a major sub-field within the disciplines, it
is expected that the Handbook will have considerable influence in the field for many years to come. Written by world authorities in philosophical logic, the work reflects careful and fruitful
collaboration by the authors at every stage of the project. This has ensured a comprehensive and definitive set of articles which will be of inestimable value to general philosophers, linguists,
logicians, mathematicians and computer scientists. Volume I: Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic -
both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of
constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the
algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume II: Extensions of Classical Logic, surveys the most significant `intensional' extensions of predicate logic
and their applications to various philosophical fields of inquiry. The twelve chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of
the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume III: Alternatives to Classical Logic, consists of a series
of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic,
quantum logic, and intuitionism. Volume IV: Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language.
Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some
criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse,
these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.
E-Book Information
• Series: Synthese Library, 164
• Year: 1,983
• Edition: 1st
• Pages: 511
• Pages In File: 511
• Language: English
• Topic: 130
• Library: Kolxo3
• Issue: 14
• Identifier: 9789400970687,9789400970663,9027715424,9789027715425
• Ddc: 160 s,160
• Lcc: BC6 .H36 1983 vol. 1,BC71 .H36 1983 vol. 1
• Commentary: wry scan (p.129,131 and most of odd pages); missing p.1-2
• Dpi: 600
• Cleaned: 1
• Org File Size: 6,018,751
• Extension: djvu
• Generic: 9bcae40009035fd424f16bed2c01ab07
• Toc:
Front Matter....Pages i-xiii
Elementary Predicate Logic....Pages 1-131
Systems of Deduction....Pages 133-188
Alternatives to Standard First-Order Semantics....Pages 189-274
Higher-Order Logic....Pages 275-329
Predicative Logics....Pages 331-407
Algorithms and Decision Problems: A Crash Course in Recursion Theory....Pages 409-478
Back Matter....Pages 479-497
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Brussels-London 9: gauge theory
University College London, 03/11/2016
Peter Kronheimer
“Instantons and the four colour problem”
Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional Z/2 vector space. The main result about this instanton homology of graphs is
a non-vanishing theorem, proved using techniques from 3-dimensional topology: if the graph is bridgeless, then its instanton homology is non-zero. It is not unreasonable to conjecture that, if the
graph lies in the plane, then the dimension of its instanton homology is equal to the number of Tait colorings of the graph (essentially the same as four-colorings of the planar regions that the
planar graph defines). If the conjecture were to hold, then the non-vanishing theorem for instanton homology would imply the four-color theorem and would provide a human-readable proof. There is some
evidence for the conjecture. This program is joint work with Tom Mrowka.
Roger Bielawski.
“Nahm’s equations and transverse Hilbert schemes”
A construction of Atiyah and Hitchin produces hyperkähler metrics on open subsets of Hilbert schemes of points on a gravitational instanton (viewed as a complex surface). The resulting hyperkähler
metrics are not the ones obtained from the Beauville-type construction on the full Hilbert scheme of points. Since the Atiyah-Hitchin construction is on the level of twistor spaces, the question of
completeness of resulting metrics requires a different approach. I’ll show how to realise these “transverse Hilbert schemes” of points on several ALF spaces as moduli spaces of Nahm’s equations,
which in particular proves completeness of the resulting metrics.
Andriy Haydys
“Topology of the blow up set for the Seiberg–Witten monopoles with multiple spinors”
A sequence of the Seiberg-Witten monopoles with multiple spinors on a three-manifold can converge after a suitable rescaling to a Fueter section, say I, only on the complement of a subset Z. I will
discuss the following question: Which pairs (I,Z) can (or can not) appear as the limit of a sequence of the Seiberg–Witten monopoles?
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What is the ROC Curve? | Data Basecamp
The receiver operating characteristic curve, or ROC curve for short, is a popular metric in machine learning for evaluating the quality of classification models. It can be used to graphically compare
different threshold values and their impact on overall performance. Specifically, it compares the trade-off between the True Positive Rate (TPR) and the False Positive Rate (FPR) at different
thresholds. In this article, we want to take a closer look at the ROC curve and better understand its interpretation and applications.
How to judge a Classification?
In the simplest case, a classification consists of two states. Suppose we want to investigate how well Corona tests reflect the infection status of a patient. In this case, the Corona test serves as
a classifier of a total of two states: infected or non-infected.
These two classes can result in a total of four states, depending on whether the classification of the test was correct:
• True Positive: The rapid test classifies the person as infected and a subsequent PCR test confirms this result. Thus, the rapid test was correct.
• False Positive: The rapid test is positive for a person, but a subsequent PCR test shows that the person is not infected, i.e. negative.
• True Negative: The rapid test is negative and the person is not infected.
• False Negative: The Corona rapid test classifies the tested person as healthy, i.e. negative, however, the person is infected and should therefore have a positive rapid test.
What is the ROC Curve and how to interpret it?
The ROC curve is a graphical representation for assessing the quality of a classifier. For this purpose, a two-dimensional diagram is created that plots the true positive rate on the y-axis and the
false positive rate on the x-axis. An existing classification model is then used and various threshold values are tested. A threshold value specifies the probability at which an instance is evaluated
as positive or negative. For example, a model that decides whether patients are classified as sick or healthy does not directly return two classes, but a probability, such as 0.67. The threshold
value decides at which of these probabilities the patient is really classified as sick.
The compromise is that a high threshold value ensures that the true positive rate is rather low, as the very certain cases are really classified as sick. However, the high threshold value also means
that potentially ill people are not detected as they are below the threshold. In the same way, the false positive rate is also rather low, as the number of false negatives is higher, which results in
a lower false positive rate.
To create the curve, various threshold values are tested and the corresponding rates are plotted. This results in an exponentially increasing curve. The perfect model would also have a point in the
top left-hand corner. This is the point with a true positive rate of 1 and a false positive rate of 0.
Example of a ROC Curve | Source: Author
In addition, a diagonal line from bottom left to top right is entered in the diagram, as this represents a completely random decision between positive and negative, which is always 50% correct. In
reality, the curves of models are somewhere between this line and the top left-hand corner. The steeper the curve and the closer it is to the upper left-hand corner, the better the model is to be
It is important to note that this graph does not make any statement about predictions of individual data points, but merely represents an overall impression of the model that makes it comparable with
other models. Other metrics such as precision, recall or F1 score should be used in addition to get a more general picture of the model’s performance.
What is the Area under the Curve?
The ROC curve is a good graph to show the trade-off between the true positive rate and the false positive rate of a model. The curves of individual classifications can also be compared by plotting
them on a common graph. However, a quantification to summarize the measure of this graph is still missing for good comparability.
In order to have a value for the classification performance, the area under the curve (AUC) is calculated. This area can have a value between 0 and 1, with a value of 1 representing a perfect
classifier. The larger the area under the curve, the better the model. Mathematically speaking, this area indicates the probability that the model classifies a random, positive instance higher than a
randomly selected negative instance.
A completely randomized model that would assign each instance after a coin toss would have an AUC value of 0.5 (the area under the red dashed line). So if a model achieves a value below 0.5, it is
worse than a random assignment of instances. An AUC value above 0.5 means that the trained model is better than a random decision. A value close to 1 indicates that the classifier can already
distinguish very well between positive and negative instances.
The advantage of the AUC value as a performance measure is that it is not dependent on the class distribution in the data set and the selection of the threshold value. In addition, this key figure
offers a good opportunity to illustrate the performance of the model by interpreting it as a probability. For example, an AUC value of 0.7 can be interpreted to mean that the model rates a random
positive instance higher than a random negative instance with a probability of 70%.
How can you use the ROC Curve in classifications with multiple categories?
The ROC curve as we have seen it so far is defined exclusively for binary classifications. In practice, however, there are so-called multi-class problems in which a model has to learn to assign a
data point to one of several classes. In order to still be able to use the ROC curve, the approach must be changed to a binary situation.
To do this, one class is taken out and defined as a positive class, the remaining classes are summarized as negative classes. This is known as a one-versus-all (OVA) approach. The ROC curve can then
be calculated for each of the cases in which one of the classes is the positive class. These curves can then be combined to form a multi-class curve.
This multi-class curve can usually be created with a so-called micro- or macro-averaging. In micro averaging, the true positive, false positive and false negative classifications are added together
for all cases and converted into a single curve. All classes are weighted equally regardless of their individual size, even if they occur with different frequencies in the initial data set. With
macro averaging, on the other hand, a separate curve is calculated for each class. The mean value is then calculated from all curves. With this approach, the classes are also weighted equally
regardless of their size.
The ROC curve is usually used for binary classifications, but can also be extended to multi-class problems using the methods described. The AUC remains a valuable measure for evaluating the overall
performance of the model and can be compared with other models for the same data set in order to find a suitable model architecture.
How does it compare to other evaluation metrics?
The ROC curve is a popular graph suitable for binary classifications. However, it is only one of many ways to evaluate classification models and it should be decided which evaluation metric is most
appropriate depending on the application. In certain scenarios, other metrics such as recall or F1-score may be more relevant.
In medical diagnosis, it is important to have a high hit rate and to recognize all positive instances, i.e. sick patients. To achieve this, it is also acceptable that a certain degree of precision is
lost and therefore some false positive errors may be included. In such cases, the ROC curve may not be optimally suited as it weights the true positive rate and the false positive rate equally.
In many applications, balanced data sets, with equal numbers of positive and negative instances, can be very difficult to create. In cases with unbalanced data, the ROC curve is not meaningful and
does not adequately reflect the performance of the model. In such applications, the precision-recall curve may be more suitable as it is optimized for unbalanced data sets.
Overall, the ROC curve is a very useful and widely used evaluation metric, but it should be adapted to the application. In addition, depending on the use case, it should also be decided whether other
metrics may be useful in order to obtain a more general picture of the model.
Can you use it in case of imbalanced datasets?
The ROC curve is a popular choice for evaluating binary classification models, but it can have problems with unbalanced data sets and give a false picture of model performance. In practice, datasets
are often unbalanced because the positive class is usually underrepresented compared to the negative class. In medical analysis, for example, datasets usually contain more healthy patients than sick
patients, or in spam detection there are often more normal emails than spam emails.
In such cases, simply looking at the accuracy of the classifier is not a good evaluation metric as it can be deceptive. Assuming a data set contains 70 % negative instances, a model that always
classifies all instances as negative can already achieve an accuracy of 70 %. Although the ROC curve gives a more honest picture here, it can still be deceptive as it focuses on both classes and can
therefore give an overly optimistic picture.
In applications with more unbalanced data sets, the precision-recall curve should therefore be used, as it concentrates exclusively on the positive class, i.e. the smaller class, and therefore
evaluates the performance of the model more independently than the ROC curve.
This is what you should take with you
• The so-called Receiver Operating Characteristic (ROC) curve serves as a graphical representation of the performance of a binary classifier.
• For this purpose, the true positive rate and the false positive rate are plotted in a two-dimensional diagram. A curve is created by plotting different threshold values.
• The threshold values determine how high the predicted probability of the model must be for an instance to be recognized as positive.
• The shape of the curve provides information about the performance. The graph of a very good model runs close to the upper left point of the diagram.
• In addition, the area under the curve (AUC) is calculated, which is a key figure for comparing different models.
• The ROC curve is originally only defined for binary classifications, but can also be extended to other applications using the so-called multi-class approach.
• The AUC value can be used as an additional assessment metric for evaluating a classification model.
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Other Articles on the Topic of ROC Curve
Scikit-Learn provides powerful functions and modules to create these measures and graphs. You can find the documentation about it here.
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Into Math Grade 3 Module 6 Lesson 1 Answer Key Represent Division
We included HMH Into Math Grade 3 Answer Key PDF Module 6 Lesson 1 Represent Division to make students experts in learning maths.
HMH Into Math Grade 3 Module 6 Lesson 1 Answer Key Represent Division
I Can use the information in a division problem to find the number of groups or the number in each group.
Spark Your Learning
Taryn works at a comic book store. She wants to put 24 comic books on display. She wants to make equal groups. How many equal groups can Taryn make? How many comic books are in each group?
Show equal groups.
Turn and Talk What is another way that Taryn can make equal groups with 24 comic books? How many equal groups would there be? How many comic books would be in each group? Explain a third way that
Taryn can make equal groups with
Given that;
Taryn works at a comic book store.
she puts comic books on the display = 24
24/6 = 4
There are 4 equal groups and each group has 6 books.
Another way is
24 = 6 × 4
Therefore multiplying 6 books and 4 groups we get 24.
And the third way is
By adding 4 groups of 6 books.
24 = 6 + 6 + 6 + 6
Build Understanding
1. Kent is making comic strips to give away at a comic book fair. He has a roll of paper that is 18 inches long. Kent draws comic strips that are each 3 inches long. How many comic strips does Kent
Show equal groups.
A. Are you finding the number in each group or the number of groups?
The number of groups is 18/3 = 6
B. How many comic strips does Kent draw?
Given that;
Kent is making comic strips band give away to a comic book fair.
The length of the paper is 18 inches long.
Kent draws each comic strip is 3 inches long
The number of comic strips done by kent is 18/3 = 6
The number of comic strips is 6.
Turn and Talk Suppose the roll of paper Kent uses is 30 inches long and Kent draws 6 comic strips. How many inches long would each comic strip be?
2. Some friends want to equally share these baseball cards. There are 5 friends. How many baseball cards does each friend get?
A. How many baseball cards are there?
The total number of baseball cards is 20.
Connect to Vocabulary
You can show a whole group separated into equal groups. To divide means to separate into equal groups.
B. Are you finding the number in each group or the number of groups?
Given that:
The total number of baseball cards is 20.
The total number of friends is 5.
Therefore 20/5 = 4
Each group has 4 baseball cards.
C. How many baseball cards does each friend get? _________
Each friend gets 4 baseball cards.
Turn and Talk How can you use what you know about multiplication to check your answer?
Check Understanding Math Board
Question 1.
Sally cuts a roll of paper into 4-inch long strips. The paper is 16 inches long. How many strips does Sally cut?
Given that;
The total length of the paper is 16 inches long.
Sallu cuts a roll of paper into 4 inches long.
The total number of strips does sally cut = 16/4 = 4.
Therefore there are 4 strips cut.
On Your Own
Question 2.
Reason Omar uses 36 craft sticks to make drink coasters. Each coaster has the same number of craft sticks. He makes 4 drink coasters. How many craft sticks does Omar use to make each coaster?
Given that;
The total number of craft sticks that Omar makes a drink coaster = 36 crafts.
The total number of drink coasters he makes = 4.
The total craft sticks does Omar use to make each coaster = 36/4 = 9
Question 3.
Keiko makes string bracelets. She has a string that is 56 inches long. Each bracelet is 8 inches long. How many bracelets does Keiko make? _________
Given that;
The total length of the string is 56 inches long.
The length of each bracelet is 8 inches long.
The number of bracelets made by Keiko make is 56/8 = 7.
Question 4.
Construct Arguments Max says 20 objects can be separated into 4 equal groups. Mara says 20 objects can be separated into 5 equal groups. Who is correct? Explain. Draw to show your answer.
I’m in a Learning Mindset!
Did my strategy for showing equal groups work? How can I adjust?
Given that;
The first statement is construct arguments max says 20 objects are separated into 4 groups.
There are 20 objects and 4 groups so, 20/4 = 5
Each group has 5 objects.
The second statement is Mara says 20 objects are separated into 5 equal groups.
There are 20 objects and 5 groups so, 20/5 = 4
Here each group has 4 objects
Therefore both the statements are true.
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Design and Analysis of Algorithms
Instructor: Martin Ziegler
Lectures: classroom #309 in building E11 (Creative Learning)
Schedule: Tuesdays and Thursdays 2:30pm to 3:45pm
Language: English
Teaching Assistants: 박세원, 이원영, 임준성, 최규현
Office hours: Tuesdays 4pm to 5:30pm and from 6:30pm, E3-1 #1434
Attendance: 10 points for missing less than 5 lectures, 9 when missing 5 lectures, 8 when missing 6, and so on: 14 or more missed lectures earn you no attendance points.
Grading: The final grade will (essentially) be composed as follows: Homework 20%, Midterm exam 30%, Final exam 40%, Attendance 10%.
≥95% for A+, ≥90% for A0, ≥85% for A-, ≥80% for B+, ≥75% for B0, ≥70% for B-, ≥65% for C+, ≥60% for C0: GPA=3.11
Exams: There will be a midterm exam on April 21 and a final exam on June 16,
both from 13h00 to 15h00 in E11 #309 (student ID ≥ 20163025) and #311 (student ID < 20163025)
Guest tutorial by Professor Neil Immerman on Descriptive Complexity on May 10 and 12 (section 5)!
PhD qualifying exam problems from Jul 1, 2016 and from Jan 13, 2017
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Model inspired by search for extraterrestrial life calculates risk of COVID-19 transmission
What are the chances of finding advanced civilizations beyond Earth? In 1961, astronomer Frank Drake developed a mathematical formula to estimate the probability of finding intelligent aliens in the
Milky Way. His simple equation, consisting of only seven variables, stimulated new discussion about an otherwise puzzling phenomenon. Decades later, his famous formula continues to influence the
search for extraterrestrial life in the universe.
Inspired by the Drake equation, fluid mechanics experts from the Johns Hopkins Whiting School of Engineering have developed a formula to answer the question of the moment: What determines someone's
chances of catching COVID-19?
In a paper published in the Physics of Fluids, the researchers present a mathematical model to estimate the risk of airborne transmission of COVID-19. Insights from this new model could help assess
how well preventive efforts, like mask wearing and social distancing, are protecting us in different transmission scenarios.
"There's still much confusion about the transmission pathways of COVID-19. This is partly because there is no common 'language' that makes it easy to understand the risk factors involved," says Rajat
Mittal, co-author of the paper and a professor in the Department of Mechanical Engineering. "What really needs to happen for one to get infected? If we can visualize this process more clearly and in
a quantitative manner, we can make better decisions about which activities to resume and which to avoid."
What's becoming clear is that COVID-19 is most commonly spread from person to person through the air, via small respiratory droplets generated by coughing, sneezing, talking, or breathing, according
to a commentary published by 239 scientists in Clinical Infectious Diseases.
But the risk of getting infected with COVID-19 depends heavily on the circumstances, Mittal says. The team's model considers 10 transmission variables, including the breathing rates of the infected
and noninfected persons, the number of virus-carrying droplets expelled, the surrounding environment, and the exposure time. Multiplied together, these variables yield a calculation of the
possibility that an individual will be infected with COVID-19.
The proposed formula is called the Contagion Airborne Transmission inequality, or CAT inequality for short.
"The CAT inequality is particularly useful because it translates the complex fluid dynamical transport process into a string of simple terms that is easy to understand," says Charles Meneveau, a
professor in Mechanical Engineering and co-author of the study. "As we've seen, communicating science clearly is of paramount importance in public health and environmental crises like the one we are
facing now."
Depending on the scenario, the risk prediction from the CAT inequality can vary greatly. Take the gym, for example. We've all heard that exercising indoors at a gym can increase your chances of
getting COVID-19, but how risky is it really?
"The CAT inequality is particularly useful because it translates the complex fluid dynamical transport process into a string of simple terms that is easy to understand. As we've seen, communicating
science clearly is of paramount importance in public health and environmental crises like the one we are facing now."
Charles Meneveau
"Imagine two people on treadmills at the gym; both are breathing harder than normal. The infected person is expelling more droplets, and the noninfected person is inhaling more droplets. In that
confined space, the risk of transmission increases by a factor of 200," Mittal says.
The team adds that the model can be useful to quantify the value of mask wearing and social distancing. If both people are wearing N95 masks, the risk of transmission is reduced by a factor of
400—that's less than a 1% chance of getting the virus. But even a simple cloth mask will significantly reduce transmission probability, according to the model. The team also found that social
distancing has a linear correlation to risk; if you double the distance, you double the protection factor, or reduce your risk by half.
As with most COVID-19 models, some variables are known and some are still a mystery. For example, we still don't know how many inhaled SARS-CoV-2 virus particles are needed to trigger an infection.
Environmental variables, like wind or HVAC systems, are also tricky to pin down.
Even with these uncertainties, the researchers believe that their model provides a useful framework for understanding how our choices can increase or reduce our risk of getting the virus. Infectious
disease models are usually designed to be understood by experts. The model developed by the team, on the other hand, is accessible to everyone, from scientists and policymakers to the average person
trying to assess their own risk.
The team hopes that taking a simple mathematical approach to a complex problem will spur new conversations about COVID-19 transmission, just as the Drake model inspired new searches for intelligent
alien life.
"With more information, you can calculate a very specific risk. More generally, our goal is to present how all these variables interact in the transmission process," Mittal says. "We think our model
can inform future studies that will close these gaps in our understanding about COVID-19 and provide better quantification of all the variables involved in our model."
Wen Wu, an assistant professor of mechanical engineering at the University of Mississippi, is a co-author of this study.
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Paper accepted to PGM 2020 | Constraint Reasoning and Optimization | University of Helsinki
The paper Learning Optimal Cyclic Causal Graphs from Interventional Data by Kari Rantanen, Antti Hyttinen, and Matti Järvisalo has been accepted for publications in the proceedings of the 10th
International Conference on Probabilistic Graphical Models (PGM 2020).
The work considers causal discovery in a very general setting involving non-linearities, cycles and several experimental datasets in which only a subset of variables are recorded. Recent approaches
combining constraint-based causal discovery, weighted independence constraints and exact optimization have shown improved accuracy. However, they have mainly focused on the d-separation criterion,
which is theoretically correct only under strong assumptions such as linearity or acyclicity. The more recently introduced sigma-separation criterion for statistical independence enables
constraint-based causal discovery for both non-linear relations and cyclic structures. The paper makes several contributions in this setting. (i) Generalizes bcause, a recent exact branch-and- bound
causal discovery approach, to this setting, integrating support for the sigma-separation criterion and several interventional datasets. (ii) Empirically analyzes different schemes for weighting
independence constraints in terms of accuracy and runtimes of bcause. (iii) Provides improvements to a previous answer set programming (ASP) based approach for causal discovery employing the
sigma-separation criterion, and empirically evaluates bcause and the refined ASP-approach.
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How many tangents can be drawn to a circle from a point on the same circle? Justify your answer.
Only one tangent can be drawn to a circle from a point on the same circle, we will prove this by constructing a line perpendicular to the radius and will prove that all other points of the line lie
exterior of the circle and hence touches the circle at a single point.
Complete step by step solution:
Suppose a circle with centre ‘O’ and a point ‘P’ on the circle and make a line perpendicular to OP, let say a line(m).
Now take any point ‘R’ on the line m,
We can say that${\text{OP < OR}}$ since the perpendicular distance is the shortest, we can say that ‘R’ is an exterior point and lie outside the circle and this will be true for all points lying on
line(m) except ‘P’.
Since line(m) touches the circle at a single point, we can say that tangent at any point of a circle is unique.
We can also prove that be the statement that tangent at any point shows the slope of that point as regarding the positive x-axis. Since the slope of a line can never have different values so tangent
to a point of a circle is unique.
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Answers created by Tanish J.
• Back to user's profile
• What is #f(x) = int 3x^3-2x+xe^(x-2) dx# if #f(1) = 3 #?
• The probability of an event is 4/5. How do you find the odds against this event?
• How do you find the x and y intercepts of #2y-5=0#?
• Question #2e9b5
• How do you factor #81p^2 - 144y^2#?
• How do you differentiate #y = x^3-8#?
• What is the mass of one mole of water?
• Question #302c7
• Question #4692d
• What is the electron configuration of Au+?
• What is the distance between #(-4,-3,4)# and #(-5,5,-6)#?
• Question #ec230
• Question #36a13
• What is the gravitational potential energy of a # 11 kg# object on a shelf # 6/5 m # high?
• If a current of #4 A# passing through a circuit generates #12 W# of power, what is the resistance of the circuit?
• How much work does it take to raise a #52 kg # weight #21 m #?
• How would you find the length of the arc on a circle with a radius of 15 cm intercepted by a central angle of 60°?
• How do you write 5,001,000 in scientific notation?
• What is #(-23pi)/12 # radians in degrees?
• How do you factor the expression #-4x^2 + 10x + 24#?
• What is the order of magnitude of 500,000?
• How do you factor the expression #4x^2 + 16x + 15#?
• How do you solve the following linear system: # 7x+2y=1 , 3x+y=3 #?
• How do you solve the following system: #3x + 2y = 1, 2x + 3y = 12#?
• What is the gravitational potential energy of a # 29 kg# object on a shelf # 5/6 m # high?
• A speed boat moving at a velocity of 25 m/s runs out of gas and drifts to a stop 3 minutes later 100 meters away. What is its rate of deceleration?
• Why do acids and bases both conduct electricity?
• The asteroid Ceres has a mass of #7*10^20# #kg# and a radius of #500# #km#. What is #g# on the surface of Ceres? How much would a #99# #kg# astronaut weigh on Ceres?
• A radioactive isotope has a half-life of 9 hours. How do you find the amount of the isotope left from a 80-milligram sample after 54 hours?
• An object is thrown vertically from a height of #3 m# at #18 m/s#. How long will it take for the object to hit the ground?
• How do you find the value of #cot 300^@#?
• Question #dbb0c
• What is #5^0#?
• How do you write 5 into an improper fraction?
• What is the x and y intercept of #x-y=5#?
• Question #d6b18
• How do you multiply expressions written in scientific notation?
• How to calculate pKa of #HCl#?
• How do you calculate slope from a graph?
• How do you graph an inequality on a number line?
• What is the price of one extension cord if Max can buy a package of 6 for $7.26?
• How do you find all the missing angles, if you know one of the acute angles of a right triangle?
• How do you write a function rule that describes a cost of a CD that is $2 each?
• How do you solve mixture problems using system of equations?
• Question #b736b
• How would you solve the following using a table? Andrew cashes a $180 check and wants the money in $10 and $20 bills. The bank teller gives him 12 bills. How many of each kind of bill does he
• How are the 6 basic trigonometric functions related to right triangles?
• How do you write a product of a number and 2 as an expression?
• How do you evaluate expressions when you have more than one variables?
• How do you evaluate #x^2+2x-1# when #x=2#?
• Question #f2bd1
• What is the reciprocal function?
• How do you cross multiply #\frac{4x}{x+2}=\frac{5}{9}#?
• How do you find the midpoint of (1.8, –3.4) and (–0.4, 1.4)?
• Why is the coordinate plane called cartesian?
• How do you create a table and graph the equation #y=2x-1#?
• What is the Zeroth law of thermodynamics?
• What is cross multiplication?
• How do you multiply #(3xy^5)(-6x^4y^2)#?
• Question #de9c1
• Question #b6ee8
• Question #b4da9
• Question #f4ee7
• Question #f81d1
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Insert A Node At Middle In Linked List using C++ - TechDecode Tutorials
Insert A Node At Middle In Linked List using C++
Technology and generation are growing up together, and the younger generation users are somewhat connected with tech and the internet all the time. Not to mention, today the whole world in this time
of crisis is working over the internet. But to make these technologies, software, etc a software developer must have excellent problem-solving skills. In this world where the internet is the new
fuel, one needs to be pretty sharp. And by sharp, for software developers, it means knowing how to automate real-world problems using computer programs. Data structures help a lot in this journey of
logic building. So today we’re going to write a simple data structure program to Insert A Node At the Middle In Linked List using C++.
What is A Linked List?
A linked list is a basic list of nodes containing different elements for all algorithms in data structures. Also, it’s pretty easy to understand.
What’s The Approach?
• Before inserting a new node we’ll first check whether the previous node is null or not. If the previous node is null then we’ll not add the new node.
• If the previous node is not null then we’ll simply create a new node and insert data into it.
• Now we’ll store the address which is stored in the previous node into this new node.
• At last, in the previous node’s address section we will store the address of this new node.
Also Read: Print Smallest Of Three Without Using Comparison Operator in C++
C++ Program To Insert A Node At Middle In Linked List
Created Linked List is: 7 8 6
// C++ program to demonstrate
// insertion method on Linked List in the Middle
#include <bits/stdc++.h>
using namespace std;
// A linked list node
class Node
int data;
Node *next;
/* Given a reference (pointer to pointer)
to the head of a list and an int, inserts
a new node on the front of the list. */
void push(Node** head_ref, int new_data)
/* 1. allocate node */
Node* new_node = new Node();
/* 2. put in the data */
new_node->data = new_data;
/* 3. Make next of new node as head */
new_node->next = (*head_ref);
/* 4. move the head to point to the new node */
(*head_ref) = new_node;
/* Given a node prev_node, insert a new node after the given
prev_node */
void insertAfter(Node* prev_node, int new_data)
/*1. check if the given prev_node is NULL */
if (prev_node == NULL)
cout<<"the given previous node cannot be NULL";
/* 2. allocate new node */
Node* new_node = new Node();
/* 3. put in the data */
new_node->data = new_data;
/* 4. Make next of new node as next of prev_node */
new_node->next = prev_node->next;
/* 5. move the next of prev_node as new_node */
prev_node->next = new_node;
/* Given a reference (pointer to pointer) to the head
of a list and an int, appends a new node at the end */
void append(Node** head_ref, int new_data)
/* 1. allocate node */
Node* new_node = new Node();
Node *last = *head_ref; /* used in step 5*/
/* 2. put in the data */
new_node->data = new_data;
/* 3. This new node is going to be
the last node, so make next of
it as NULL*/
new_node->next = NULL;
/* 4. If the Linked List is empty,
then make the new node as head */
if (*head_ref == NULL)
*head_ref = new_node;
/* 5. Else traverse till the last node */
while (last->next != NULL)
last = last->next;
/* 6. Change the next of last node */
last->next = new_node;
// This function prints contents of
// linked list starting from head
void printList(Node *node)
while (node != NULL)
cout<<" "<<node->data;
node = node->next;
/* Driver code*/
int main()
/* Start with the empty list */
Node* head = NULL;
// Insert 6. So linked list becomes 6->NULL
append(&head, 6);
// Insert 7 at the beginning.
// So linked list becomes 7->6->NULL
push(&head, 7);
// Insert 8, after 7. So linked
// list becomes 7->8->6->NULL
insertAfter(head, 8);
cout<<"Created Linked list is: ";
return 0;
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Can slope be negative?
Asked by: Verner Gibson Score: 4.9/5
46 votes
A negative slope means that two variables are negatively related; that is, when x increases, y decreases, and when x decreases, y increases. Graphically, a negative slope means that as the line on
the line graph moves from left to right, the line falls.
Is the slope always positive?
When calculating the rise of a line's slope, down is always negative and up is always positive. When calculating the run of a line's slope, right is always positive and left is always negative.
Is the slope negative or positive?
Pattern for Sign of Slope
If the line is sloping upward from left to right, so the slope is positive (+). If the line is sloping downward from left to right, so the slope is negative (-).
What does negative slope look like?
Graphically, a negative slope means that as the line on the line graph moves from left to right, the line falls. We will learn that “price” and “quantity demanded” have a negative relationship; that
is, consumers will purchase less when the price is higher. ... Graphically, the line is flat; the rise over run is zero.
How does a slope look like?
The slope equals the rise divided by the run: . You can determine the slope of a line from its graph by looking at the rise and run. One characteristic of a line is that its slope is constant all the
way along it. So, you can choose any 2 points along the graph of the line to figure out the slope.
Positive and negative slope | Algebra I | Khan Academy
39 related questions found
What is an example of a zero slope?
Zero Slope and Graphing
Just like in the bicycling example, a horizontal line goes with zero slope. One thing to be aware of when you graph, however, is that this horizontal line can be any height. For example, the picture
you see here has three horizontal lines. In each case the slope is zero.
What does a slope of 2 look like?
In other words, our line moves 2 units upward every time it moves 1 unit to the right. Our slope is 2. It's a positive number, so we rise up and run to the right. Or, if we want to be contrary, both
the rise and run could be negative, moving down and to the left.
What is the equation for a zero slope?
A zero slope line is a straight, perfectly flat line running along the horizontal axis of a Cartesian plane. The equation for a zero slope line is one where the X value may vary but the Y value will
always be constant. An equation for a zero slope line will be y = b, where the line's slope is 0 (m = 0).
What happens when the slope of the line is negative?
If the slope is negative, then the rise and the run have to be opposites of each other, one has to be positive and one has to be negative. In other words, you will be going up and to the left OR down
and to the right.
What does a negative slope mean?
Visually, a line has negative slope if it goes down and right (or up and left). Mathematically, this means that as x increases, y decreases.
Can a slope be greater than 1?
Slopes can be more than one and less than negative one. A slope determines how steep a line is and the sign indicates if it's going "uphill" or "downhill". The lowest absolute slope (the absolute
value of a slope) is 0 which means the line is perfectly horizontal.
Can there be a slope of 0?
Well you know that having a 0 in the denominator is a big no, no. This means the slope is undefined. As shown above, whenever you have a vertical line your slope is undefined.
How do you determine the slope?
Slope can be calculated as a percentage which is calculated in much the same way as the gradient. Convert the rise and run to the same units and then divide the rise by the run. Multiply this number
by 100 and you have the percentage slope.
What is the slope of 3?
y=3 would be nothing more than a horizontal line through y=3. So the rise is always 0 (it never goes up or down) and the run is always the distance from zero to any point on the line. In other words,
the slope would be 0/the change in x, which is always 0.
What is the slope of the line through (- 1 and 3 9?
The slope is 4 .
What are the correct properties of slope?
1) The slope of a line is constant and any two points on the line may be used to find its value. 2) A line with positive slope rises from left to right. 3) A line with negative slope falls from left
to right. 4) The slope of a horizontal line is equal to 0.
Can you have a slope of 0 6?
Answer and Explanation: No, the slope 06 is not undefined. By definition, an undefined slope is a slope with a 0 in the denominator of the slope.
What does a slope of infinity look like?
An infinite slope is simply a vertical line. When you plot it on a line graph, an infinite slope is any line which runs parallel to the y-axis. You can also describe this as any line that doesn't
move along the x-axis but stays fixed at one constant x-axis coordinate, making the change along the x-axis 0.
Is a slope of 0 4 undefined?
04=0 is defined. 40 is not.
What is a gentle slope?
adjective. A gentle slope or curve is not steep or severe.
How do you find the slope given two points?
Use the slope formula to find the slope of a line given the coordinates of two points on the line. The slope formula is m=(y2-y1)/(x2-x1), or the change in the y values over the change in the x
values. The coordinates of the first point represent x1 and y1. The coordinates of the second points are x2, y2.
Which number is the slope in a linear equation?
In the equation of a straight line (when the equation is written as "y = mx + b"), the slope is the number "m" that is multiplied on the x, and "b" is the y-intercept (that is, the point where the
line crosses the vertical y-axis). This useful form of the line equation is sensibly named the "slope-intercept form".
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Finding R and S for Tricky Examples - Organic Chemistry | Socratic
Finding R and S for Tricky Examples
Key Questions
• Explanation:
Step 1. Draw the structure of 1-bromo-1-chlorobutane.
Step 2. To get the stereochemical structure, start with the structure of butane.
Step 3. At $\text{C-1}$, draw a wedge, a dash, and a solid bond.
Step 4. Put the $\text{H}$ on the dashed bond. Then add the $\text{Br}$ and $\text{Cl}$ to the remaining bonds in any order.
We have a 50:50 chance of getting it right. Here's one possibility.
Step 5. Assign priorities to the groups.
$\text{Br}$ = 1; $\text{Cl}$ = 2; $\text{C-2}$ = 3; $\text{H}$ = 4.
Step 6. Assign stereochemistry
Then $\text{Br}$ → $\text{Cl}$ → $\text{C-2}$ = 1 → 2 → 3 goes in a counterclockwise direction ($S$).
We got it right!
So this really is ($S$)-1-bromo-1-chlorobutane.
If we had gotten it wrong, we would have interchanged any two of the groups.
For example, in the diagram above, interchanging the $\text{Cl}$ and $\text{Br}$ atoms converts the ($S$) isomer to the ($R$) isomer.
• Answer:
The structure is below...
Both $C l$ groups should be pointing backward. After assigning priorities, the rotation will be counter-clock wise (which means $S$) however because the highest priority group is pointing
backward, the $S$ becomes $R$.
A video on labeling $R$ and $S$ is here:
• Answer:
Because you have to represent a three dimensional structure on two dimensional paper.
This is always problematic, and you can make mistakes even with reference to molecular models. It depends how well your spatial awareness can translate the structure to the printed page. Good
Stereochemistry (R and S), Isomers, and Optical Activity
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Comparison of Calculated Fit and Experimental Calculations of Average Dose Deposited in Aluminum by High Energy Electron Beams
Journal of Modern Physics Vol.08 No.05(2017), Article ID:75434,9 pages
Comparison of Calculated Fit and Experimental Calculations of Average Dose Deposited in Aluminum by High Energy Electron Beams
Mohammad Farnush^
School of Metallurgy & Materials Engineering, College of Engineering, University of Tehran, Tehran, Iran
Copyright © 2017 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: March 11, 2017; Accepted: April 14, 2017; Published: April 17, 2017
This paper presents the formalism for absorbed dose determination to Aluminum in high-energy electron beams using Rhodotron accelerator. Depth dose curve for Aluminum at electron energy of 10 MeV was
calculated. The calculated curve in the model as a function of the depth is compared to the experimental. The agreement of the final results remained well within the expected acceptable range. The
calculated values of dose-to-Aluminum are completely fit with the measured values in the range of 0.07% for electron energy of 10 MeV.
Calculated Fit, Average Dose, Aluminum, High Energy Electron Beams
1. Introduction
Researchers presented radiation processing more than 50 years ago [1] , and they have developed many beneficial applications. The most delicate business applications relate to reforming a variance of
plastic and rubber productions, and germ-free curative devices and spent items. Appearing applications are posterior and retreat foods, and shortening provisional contamination.
Electron beam gun is an imperative constituent of Gaussian accelerators, which have a difference of applications in examination particularly in high-energy electron beams and industry. In considering
electron-positron physics, e-beam is the necessity constituent. This is roughly succeeded throughout thermionic ray of electrons though several other situations of e-beam fabrication as well pursue.
This organization of ray has two attitudes: temperature and space charge restrained ray. In first organize; the temperature of the cathode materials restrains the ray while the last ray is restrained
by the space charge about the cathode corresponding insufficient acceleration likelihood. Receiving consider its pro- ceeds and repose in fabrication, retaining exercises are observed for the
increase of high-energy thermionic electron gun for close origination Gaussian accelerators. These guns obligate fractionally low vacuum and have long liability continual. A critical passage on these
types of the guns can be rescued in the referral. In the represent study, electron beam guns used up in some of the well-under- stood high-energy physics laboratories are in short enumerated. Embargo
linear accelerator, all references under compassions is thermionic in environment. We as well represent one of our dynamically fabricated sources for conforming and explication with the being sources
[2] .
Rhodotrons are electron accelerators correlate the quality of “re-circulating” a beam through consecutive diameters of a singular coequal cavity reverberating in metric waves. Such intension makes it
expedient to attain CW acceleration of electron beams to high energies. The primary implication principle of the Rhodotron was first propound in 1989 by J. Pottier from the French Atomic Energy
Agency (CEA), who proposed the use of a half-wavelength coequal cavity, shorted at both ends to accelerate electrons. The quality of handling of the Rhodotron has already been enumerated in length in
previous articles. The first capitalist Rhodotron (mentioned as model TT-200) was produced in 1993 at IBA Belgium. It surpassed its obligated presentations in 1994 with a maximum beam output power of
110 kW at 10 MeV (100 kW contemplated), and a high power productiveness of 38% at full beam power (25% expected). Three industrialist Rhodotron systems ranging from 35 kW to 200 kW beam power at 10
MeV have been absolutely designed, and are continually constructed at IBA’s specialties in Louvain-la-Neuve. Contingencies of these capitalist accelerators were selected in order to manner the
neediness of the outlet of capitalist irradiation for simple, compressed and trustworthy high-power EB units [3] .
Electron accelerators have a wide range of requests in the industry and in the essential examination. Two of the most common issues whilst material irradiation with electron beams are determining the
absorbed dose and the reached depth. The approach in each instance adheres on the variety of irradiator, the technological resources and the fantastic usable [4] .
In the designing of high energy electron beam processing, calculation is necessary of the absorbed dose in an example under given irradiation requirements. Even for the most unimportant example
arrangement of a glide covering, the issue to be treated is the passing of electrons throughout a covering absorber containing of an accelerator window [5] .
The electron beam already has a specific performance expansion before assembly the accelerator outlet window. This expansion relies on a large number of reasons such as sort of acceleration
instrument, procedure of command and withdrawal, and beam manipulation system. The energy spectral at the accelerator outlet window (vacuity side), and may be distinguished by a comparatively mean
issue of parameters such as though verified: Emax is the outmost energy, Ea is the average electron energy, Ep, a is the most possible electron energy and Γa is the energy expansion, i.e. the
expansion at half maximum of the energy spectral [6] .
Monte Carlo simulations are intensely effective for various physical procedures. The transportation of portions was simulated by Monte Carlo calculating the essential parameters such as contingencies
of transmitter-detected and ungainly-energy scattering after interaction with material. The objective of Monte Carlo representation of electron-solid interaction is to simulate the dispersal
procedures as correctly as acceptable in the average-energy arrange [7] . This paper fit the dose curve by Monte Carlo but we are not comparing with Molecular dynamics or other methods.
Most of the theoretical accomplish for estimation the absorbed dose in an irradiated matter manipulates calculation procedures. Monte Carlo simulation and straight numerous conclusion of
transportation calculation are some useful procedures. However, both of them require enormous CPU time and make necessity the use of high volume computers. In this work, we present a procedure
applying a standard private computer for determining indicates of the depth- dose curve of mono-energetic electrons with an energy arrange of 10 MeV. The performed significant allow inside an
acceptable arrange, with the experimental estimations. This coherent consents us to have a general experimental procedure to adjust high-energy electron accelerators.
2. Materials and Method
An ionization compartment used (Rhodotron) and its characters are recorded in Table 1. The absorbed dose to Aluminum caliber reasons were supplied by a Secondary Standard Dose Laboratory (SSDL).
Complete charge was capacitated with the electroplate example.
The measurements were all performed at the Gaussian accelerator Rhodotron. In the case of high-energy electrons, the two available energies of 5 MeV and 10 MeV. The measurements and comparisons in
high-energy electron beams apply to 10 MeV. The method for extracting the depth dose curve by experiment is to put a film on aluminum to measure the dose on the base of the depth.
Basic Formalism
Radiation processing can be determined as the manipulation of maters and produces with radiation or ionizing energy to exchange their physical, chemical or biological characteristics, to enlarge
their advantage and value, or to decrease their impact on the circumstance. Accelerated electrons, X-rays (bremsstrahlung) emitted by energetic electrons, and gamma rays emitted by radioactive
nuclides are appropriate energy sources. These are all accomplished of ejection atomic electrons, which can then ionize other atoms in an overflow of impacts.
Table 1. Characteristics of Rhodotron.
So they can generate similar molecular induces. The selection of energy source is usually based on effective concepts, such as absorbed dose, dose similarity (max/ min) ratio, mater depth, density
and arrangement, computation rate, capital and managing costs [8] .
In the case of electron beam (EB) processing, the contingency electron energy establishes the maximum mater depth, and the electron beam current and power establish the maximum processing rate [9] .
3. Absorbed Dose Definition
The most essential specification for any irradiation process is the absorbed dose. The quantitative effects of the procedure are reported to this factor. Absorbed dose is symmetrical to the ionizing
energy delivered per unit mass of mater. The international unit of dose is the gray (Gy), which is determined as the absorption of one joule per kilogram (J/kg) [9] . A more accessible unit for most
radiation processing applications is the kilo gray (kJ/kg or J/g). An older unit is the rad, which is determined as the absorption of 100 ergs per gram or 10 - 5 joules per gram. So, 100 rads is
equivalent to 10 - 3 joules per gram or 1 joule per kilogram or 1 gray. The rad unit is now obsolete, but several marketable procedures are still categorized in rads, kilorads or megarads [10] .
The relationship between absorbed dose of Aluminum and electron energy deposition of Aluminum can be derived in the following manner:
Absorbed dose of Aluminum =
Absorbed energy of Aluminum =
Mass of Aluminum =
Electron Beam Power (KW) = EBP
Irradiation Time (S) = IT
Electron Energy =
Electron Beam Current =
Energy Deposition per Incident Electron per unit area Density =
Layer Thickness of Aluminum =
Volume Density of Aluminum =
Then Equation (1) can be modified as follow:
Absorbed Dose at depth z in the irradiated Aluminum =
Energy Deposition per Electron at Depth z in the irradiated Aluminum =
Fraction of Emitted Beam Current intercepted by the irradiated Aluminum =
Then Equation (6) can be modified as follow:
4. Results
The penetration of the fast electrons through the material is the respectable subject of the theoretical and experimental explorations. Multiple Coulomb dissipation and statistical uncertainty in
ionization energy loss distinguish it. The transportation attributes of electrons are also of attention in correlation with great experimental applications. When the absorber maters are thin enough
resembled to the range of contingency electrons, the problems can be disciplined analytically and the adequate coherence between theory and experiment has been reported [11] .
The condition becomes more perplexed when the absorber thickness expansions. For this case, the energy loss and ungainly deflection cannot be handled exclusively and theoretical treatment of the
problem is very puzzling. There are two procedures to study electron transport in the thick maters; the moment’s arrangement and the Monte Carlo method. The moments method progressed by Spencer’ has
been used to calculate the depth distribution of energy deposition in the maters. Nevertheless, its application is confined only to media that are released and comparable, and foil-transmission
problems cannot be handled by this procedure [11] .
On the other hand, the Monte Carlo method can, in opinion, present the most methodical resolutions for the electron transport problems in confined media. This procedure is appropriate to any energy
range of electrons and to any geometry. The calculation is based on the simulation of the electron tracks by incidental sampling techniques. The numerate of Coulomb accidents undergone by a fast
electron while the slowing-down procedure is exactly big, so that the simulation of the individual impact is not feasible. Instead, the electron trajectory is separated into a numerate of short
sections, such that the numerate of impacts along each section is big, however the average ungainly duplicated due to varied scattering and the average energy loss by ionization per section are
small. This means that the conclusions of systematic managements of angular multiples and energy losses can give adequate propinquities to the net contingency dispersions in each section. The angular
deflections and energy losses inspected in each section are then connected to fabricate a complete electron trajectory in the foil. By the use of such a method, the thick-foil transfer problem can be
handled as a consequence of thin-foil ones, in which logical forms can satisfactory present the electron performance [11] .
Berger has informed the Monte Carlo procedure based on this perspective, and the previously references are cited therein. Recently, Sugiyama’ executed Monte Carlo estimations for transportation of
fast electrons with energy higher than a few MeV and acquired the conclusions in extremely good agreement with experimental data. Berger and Seltzer progressed a computer code entitled ETRAN, (“which
the most actual and the most repeatedly used program is at present. by the use of this code they estimated disparate quantities for electron transmittal”) and for construction of bremsstrahlung
radiation. Using the same intellect as those manipulated in the ETRAN code, many codes have been developed to authorize the practice studying of the electron transportation problems for
one-dimensional multilayer objectives (“cylindrical-geometry multi-maters’ and combined procedures [11] ”).
Regardless the ETRAN code is actual, takes into consider several interactions between electron and atom, and has the flexibility of being able to estimate disparate quantities, it obligates a
large-memory and high-speed computer. In the case of radioactive sources, most electrons speeded from the source have the energy less than 2 MeV [11] .
In this paper is presented a brief narration of a Calculated Fit program to study transmittal of electrons through Aluminum. This program comprehends electron complicated scattering and the
exestuation of energy-loss straggling. The usefulness is its clearness and high speed. The present code does not obligate large-memory computer and is adequately actual. Estimations have been
conveyed with this code for mono-energetic electrons incident perpendicularly on the objectives of different depth. The consequences presented here contain the transmission coefficient and the energy
spectral of transmitted electrons. Proportion is made with the experimental conclusions and also with the estimated result from the Calculated Fit code [11] .
A fundamental problem in the dosimetry of electrons is the estimation of the average absorbed dose disclosed to a mater suspicion perpendicular by a homogeneous plane-parallel broad beam of
mono-energetics electrons. Yet the complications of scattering, straggling, and the production and escape of delta-rays and bremsstrahlung may cause inaccuracies in simple analytical calculations
using CSDA (continuous slowing down approximation) stopping-power or range tables such as those of Berger and Seltzer (1983) [11] .
In the present paper analytical calculations of dose that depend on various conjectures about the complication processes are carried out, and are compared with corresponding Calculated Fit
computations which provide the quantities shown in Figure 1 [11] .
To compare Calculated Fit with dose determinations, a model of a flat sheet of aluminum moving continuously through a wide electron beam was used. The quantities that must be known are:
1) The contingency electron energy,
2) The accelerated beam current,
3) The fraction of beam current overtook by the irradiated mater,
Figure 1. Depth dose curve for aluminum at electron energy of 10 MeV. The calculated curve in the model (__) as a function of the depth is compared to the experimental (.).
4) The atomic composition and depth of the beam window,
5) The air space between the beam window and the irradiated mater,
6) The dosimeters and any covering or supporting maters,
7) The composition, width and thickness of the irradiated mater and
8) The conveyor speed. The Calculated Fit code calculates the Dose,
4.1. Method of Calculation
The Calculated Fit has been calculated under the following assumptions:
1) An electron beam is incident normally on aluminum.
2) The lateral extension of the aluminum is large enough compared to the electron range.
3) The electrons never return to the aluminum once they have left it.
4) The production of knock-on electrons by the inelastic Coulomb collisions is ignored.
5) The energy loss by radiation is neglected [11] .
4.2. Analytical Calculation
For this simplest case we make the assumptions:
1) The electrons that enter the Aluminum pass straight through without scattering.
2) The stopping power value for the incident energy r, applies throughout the electron path length, but the maximum energy spent in the Aluminum is limited to T,
3) Any bremsstrahlung that is produced escapes.
4) Delta-ray effects are negligible.
5) Electron backscattering is negligible [11] .
The results obtained from the Calculated Fit in the model for the Dose-Depth distribution and experimental are given in Figure 1. The depth-dose distribution given by the present algorithm has been
compared with experimental.
Smoothing spline:
Dose =
Smoothing parameter:
P = 0.99992527
Goodness of fit:
SSE: 0.007257
R-square: 0.9996
Adjusted R-square: 0.9988
RMSE: 0.03167
5. Discussion
In the characterization of the electron beam, it is usable to declare some specialty of the procedure represented here. Firstly, it is very simple and it allows us to have an average value of the
current density. However, from the digitized spots of the beam of our system it is possible to infer the arrangement profile.
The results of absorbed dose according to Aluminum obtained with the one chamber was apparently highly consistent, if the result with the Rhodotron chamber at 10 MeV is included. The good agreement,
however, is simply a result of the fact that equation 8 was used to derive appropriate [11] .
6. Conclusions
The calculated values of dose-to-Aluminum are completely fit with the measured values in the range of 0.07% for electron energy of 10 MeV. Possible explanations for these small differences have not
been investigated. Even so, the relatively close agreement is satisfying when the extreme differences in dose rate (10000/1) and type of radiation (electrons versus gamma rays) are considered. This
study confirms that the response of this type of dosimeter system is independent of the dose-rate, and it provides assurance that Calculated Fit calculations can give results with sufficient accuracy
for many industrial applications of radiation processing [11] .
The present approach describes the penetration of the primary electrons for aluminum and with considerable success. A comparatively simple model gives a reasonable description of electron scattering
for energy of 10 MeV. The scattering processes involve elastic and inelastic scattering. The calculation provided the energy spectra and angular distributions of transmitted and reflected electrons
for aluminum by Calculated Fit approach. Fitting results for transmission experiments are presented and compared with experimental data for electron energy of 10 MeV. Such a Calculated Fit procedure
can be efficiently used to fit the experimental conditions encountered in surface electron spectroscopy [11] .
We acknowledge technical and scientific help of University of Tehran.
Cite this paper
Farnush, M. (2017) Comparison of Calculated Fit and Experimental Calculations of Average Dose Deposited in Aluminum by High Energy Electron Beams. Journal of Modern Physics, 8, 747-755. https://
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About Mitchel T. Keller
Author BiographyAbout Mitchel T. Keller
Mitchel T. Keller is a super-achiever (this description is written by WTT) extraordinaire from North Dakota. As a graduate student at Georgia Tech, he won a lengthy list of honors and awards,
including a VIGRE Graduate Fellowship, an IMPACT Scholarship, a John R. Festa Fellowship and the 2009 Price Research Award. Mitch is a natural leader and was elected President (and Vice President) of
the Georgia Tech Graduate Student Government Association, roles in which he served with distinction. Indeed, after completing his terms, his student colleagues voted to establish a continuing award
for distinguished leadership, to be named the Mitchel T. Keller award, with Mitch as the first recipient. Very few graduate students win awards in the first place, but Mitch is the only one I know
who has an award named after them.
Mitch is also a gifted teacher of mathematics, receiving the prestigious Georgia Tech 2008 Outstanding Teacher Award, a campus-wide competition. He is quick to experiment with the latest approaches
to teaching mathematics, adopting what works for him while refining and polishing things along the way. He really understands the literature behind active learning and the principles of engaging
students in the learning process. Mitch has even taught his more senior (some say ancient) co-author a thing or two and got him to try personal response systems in a large calculus section.
Mitch is off to a fast start in his own research career, and is already an expert in the subject of linear discrepancy. Mitch has also made substantive contributions to a topic known as Stanley
depth, which is right at the boundary of combinatorial mathematics and algebraic combinatorics.
After finishing his Ph.D., Mitch received another signal honor, a Marshall Sherfield Postdoctoral Fellowship and spent two years at the London School of Economics. He is presently an Assistant
Professor of Mathematics at Washington and Lee University, and a few years down the road, he'll probably be president of something.
On the personal side, Mitch is the keeper of the Mathematics Genealogy Project, and he is a great cook. His desserts are to die for.
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Polar Coordinates and Graphs
I remember being frightened by the polar coordinate system and graphing equations using it when I was in my students’ seats many years ago. With graphing calculators, this doesn’t have to be
I started the unit with an introduction to the polar coordinate system and reminded my students about how intimately familiar they are with the unit circle and how closely this resembles it. I passed
out the Vizual Notes and we learned how to plot points, name a point in three additional ways, and how to convert between the two systems. Then my students practiced using their new knowledge.
Next we discussed how to convert rectangular equations to polar and vice versa.
Then it’s on to an investigation to see how much my students can learn by graphing polar equations using a graphing calculator.
Leave A Comment
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An Improved Algorithm for the Solution of Generalized Burger-Fishers Equation
An Improved Algorithm for the Solution of Generalized Burger-Fishers Equation ()
1. Introduction
Generalized Burger-Fisher equation, being a nonlinear partial differential equation, is of great importance for describing the interaction between reaction mechanisms, convection effects, and
diffusion transports. Since there exists no general technique for finding analytical solution of nonlinear diffusion equations so far, numerical solutions of nonlinear equations are of great
importance in physical problems.
Many researchers [1] -[13] have used various numerical methods to solve Generalized Burger-Fisher. Recently Javidi [10] used modified pseudospectral method for generalized Burger’s-Fisher equation.
Kaya [2] introduced a numerical simulation of the generalized Burger’s-Fisher equation. Ismail [8] presented a restructive pade approximation for the solution of the generalized Burger’s-Fisher
equation. Hassan et al. [3] studied Adomian Decomposition Method (ADM) for generalized Burger’s-Huxley and Burger’s-Fisher equations.
Unlike some previous methods that used various transformations and several iterations, we present a new Modified Variational Iteration Method (MVIM) for the numerical solutions of generalized
Burger-Fisher equation.
In Table 1, the results of MVIM were compared with those of ADM and VIM when
In Table 2, we compared MVIM, ADM and VIM results for Table 3 the results are compared for
Figure 1 and Figure 2 show the graphical representation of gBF for various values of
Figure 3 shows the graph of exact solution and MVIM solution. Figure 4 also represents graph of gBF when
2. Modified Variational Iteration Method (MVIM)
The idea of variational iteration can be traced to Inokuti [9] . The variational iteration method was proposed by J.-H. He [4] - [7] , In this paper, a Modified Variational Iteration Method proposed
by Olayiwola [11] - [14] is
presented for the solution of the generalized Burger-Fisher equation.
To illustrate the basic concept of the MVIM, we consider the following general nonlinear partial differential equation:
where L is a linear time derivative operator, R is a linear operator which has partial derivative with respect to
3. MVIM for the Solution of Generalized Burger-Fisher Equation
The following generalized Burger-Fisher (gBF) equation problems arising in various field of science is considered.
with the initial condition
And the boundary conditions
We used Maple to code (1.1 - 1.2) for the solution of (1.3 - 1.6) and the following results were obtained after one iteration:
When Table 3.
4. Results and Discussion
Tables 1-3 shows that the MVIM is the best approximant when compared with VIM and ADM. Figure 2 is the graph of Exact solution for the generalized Burger-fisher when
Figure 3 compares the graph of Exact with the MVIM. It is also to be noted that both graphs of Burger-Fisher and generalized Burger-Fisher as shown in Figure 1 and Figure 3, respectively, justify the
conclusion that the two equations approaches the same steady state. However, as Figure 4.
5. Conclusions
There are some important points to note here. First, the MVIM provides the solutions in terms of convergent series with easily computable components. Second, it is clear and remarkable that
approximate solutions using MVIM are in good agreement. Third, the MVIM technique requires less computational work than many existing approaches. The MVIM was used in a direct way without using
linearization, perturbation or restrictive assumptions.
The MVIM provides more realistic series solutions, very high accuracy, fast transformation and possibility of implementation of algorithm. The Algorithm makes it easier for the system to predict the
next series.
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early payoff calculator auto loan
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of the money. The document evidencing the debt (e.g., a promissory note) will normally specify, among other things, the principal amount of money ...
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Using SQL aggregate functions with a LEFT JOIN
I've got two datasets. The first dataset (DATA_WITH_MANY_ROWS) is a dataset with many rows of data and includes two numeric variables numvar1 and numvar2. Using only one PROC SQL step, I would like
to 1) aggregate the data across this entire table, then 2) aggregate the data within a subset of the table, then 3) join with an existing dataset that has only one observation (DATA_WITH_ONE_ROW).
You'll also notice I'm including adding a text label to the data. Essentially I am trying to export a final dataset which contains only one observation and side by side includes the above described
I have included code below showing my current approach. However, the output table (OUTPUT_TABLE) ends up having as many rows as the dataset DATA_WITH_MANY_ROWS. I can't seem to figure out where my
code is going wrong since I'm fairly certain each item in the left-join should only have one observation and should join smoothly using the ON 1=1 condition. Since I am summarizing in every case down
to a single row of data, I assume I do not need to use group by functions.
Am I misunderstanding something about how the order that SQL is processing my code? Thanks to the community for any help/insight.
PROC SQL;
CREATE TABLE work.OUTPUT_TABLE AS
"texthere" AS label_everyone,
SUM(a.numvar1) AS TOTAL1 SUM(a.numvar2) AS TOTAL2,
b.*, c.*
FROM work.DATA_WITH_MANY_ROWS AS a
(SELECT SUM(numvar1) AS TOTAL1_SUBSET, SUM(numvar2) AS TOTAL2_SUBSET FROM DATA_WITH_MANY_ROWS WHERE classvar=1) AS b ON 1=1
LEFT JOIN DATA_WITH_ONE_ROW AS c ON 1=1;
01-28-2016 03:38 AM
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004- USMA(WP) entrance exam question independent variable
004- USMA(WP) entrance exam question independent variable
• MHB
• Thread starter karush
• Start date
In summary, the interval of possible values for the dependent variable is [8, 72] and for positive values of \theta, \theta^2, and 2\theta^2 is an increasing function. The values of 2\theta^2, for x
between 2 and 6, lie between 8 and 72. The problem does not specify the measurement unit for \theta and it is simply treated as a number.
ok not sure what forum this was supposed to go in,,so...
If the independent variable of $W(\theta)=2\theta^2$ is restricted to values in the interval [2,6]
What is the interval of all possible values of the dependents variable?
Last edited:
For positive values of [tex]\theta[/tex], [tex]\theta^2[/tex], and therefore [tex]2\theta^2[/tex], is an increasing function. Values of [tex]2\theta^2[/tex], for x between 2 and 6, lie between [tex]2
(2^2)= 8[/tex] and [tex]2(6^2)= 72[/tex].
HallsofIvy said:
For positive values of [tex]\theta[/tex], [tex]\theta^2[/tex], and therefore [tex]2\theta^2[/tex], is an increasing function. Values of [tex]2\theta^2[/tex], for x between 2 and 6, lie between
[tex]2(2^2)= 8[/tex] and [tex]2(6^2)= 72[/tex].
Ok I think I got ? Because $\theta$ ussually means radians or degrees!
Nothing is said, in this problem, about [tex]\theta[/tex] being the measure of an angle or any measurement at all. It's just a number.
FAQ: 004- USMA(WP) entrance exam question independent variable
1. What is the purpose of the independent variable in the USMA(WP) entrance exam question?
The independent variable is the factor that is being manipulated or changed in an experiment in order to observe its effect on the dependent variable. In the USMA(WP) entrance exam question, the
independent variable is used to test the candidate's ability to think critically and solve problems.
2. How is the independent variable chosen for the USMA(WP) entrance exam question?
The independent variable is carefully selected based on the specific skills and knowledge that the USMA(WP) is looking for in potential candidates. It is often chosen to reflect real-world situations
and challenges that the candidates may face in their future military careers.
3. Can the independent variable change throughout the USMA(WP) entrance exam question?
No, the independent variable is typically kept constant throughout the entire exam question in order to accurately measure the candidate's performance. Changing the independent variable during the
exam could introduce confounding variables and affect the validity of the results.
4. How does the independent variable affect the overall outcome of the USMA(WP) entrance exam question?
The independent variable plays a crucial role in determining the outcome of the USMA(WP) entrance exam question. It is the variable that is being tested and evaluated, and its effect on the dependent
variable is used to assess the candidate's abilities and potential for success in the military.
5. Can the independent variable be controlled by the candidate during the USMA(WP) entrance exam question?
No, the independent variable is predetermined and cannot be controlled by the candidate. This is to ensure that all candidates are given the same conditions and opportunities to demonstrate their
skills and abilities, and to maintain the fairness and integrity of the exam.
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Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions
We suggest here a least-change correction to available finite element (FE) solution. This postprocessing procedure is aimed at recovering the monotonicity and some other important properties that may
not be exhibited by the FE solution. It is based on solving a monotonic regression problem with some extra constraints. One of them is a linear equality-type constraint which models the
conservativity requirement. The other ones are box-type constraints, and they originate from the discrete maximum principle. The resulting postprocessing problem is a large scale quadratic
optimization problem. It is proved that the postprocessed FE solution preserves the accuracy of the discrete FE approximation. We introduce an algorithm for solving the postprocessing problem. It can
be viewed as a dual ascent method based on the Lagrangian relaxation of the equality constraint. We justify theoretically its correctness. Its efficiency is demonstrated by the presented results of
numerical experiments.
Technical Report LiTH-MAT-R-2011/08-SE, Department of Mathematics, Linköping University, Sweden, 2011
View Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions
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Rectangular Axis
Let XOX’ and YOY’ be two fixed straight lines, which meet at right angles at O. Then,
(i) X’OX is called axis of X or the X-axis or abscissa.
(ii) Y’OY is called axis of Yor the Y-axis or ordinate.
(iii) The ordered pair of real numbers (x, y) is called cartesian coordinate .
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Multiple Stacked Bar Chart Power Bi 2024 - Multiplication Chart Printable
Multiple Stacked Bar Chart Power Bi
Multiple Stacked Bar Chart Power Bi – You can create a Multiplication Chart Nightclub by marking the columns. The remaining line ought to say “1” and symbolize the amount increased by one. On the
right hand part of your kitchen table, content label the posts as “2, 6, 4 and 8 and 9”. Multiple Stacked Bar Chart Power Bi.
Suggestions to understand the 9 times multiplication table
Discovering the nine periods multiplication desk is not a simple task. Counting down is one of the easiest, although there are several ways to memorize it. Within this trick, you place your hands on
the desk and amount your hands and fingers one at a time from one to twenty. Collapse your 7th finger to enable you to see the tens and ones upon it. Then count up the quantity of hands and fingers
on the left and proper of your folded finger.
When discovering the desk, kids might be afraid of bigger amounts. This is because incorporating larger figures consistently is a task. However, you can exploit the hidden patterns to make learning
the nine times table easy. A technique would be to write the 9 periods table over a cheat page, read through it all out loud, or process composing it downward frequently. This process can make the
desk much more unique.
Habits to find over a multiplication chart
Multiplication graph or chart bars are great for memorizing multiplication information. You will discover the item of two phone numbers by exploring the rows and columns from the multiplication
graph. By way of example, a column that may be all twos along with a row that’s all eights ought to fulfill at 56. Styles to consider on a multiplication chart club are like those who are in a
multiplication table.
A design to look for on the multiplication chart may be the distributive property. This house may be noticed in all of the columns. As an example, a product or service x two is equal to 5 various
(periods) c. This identical residence applies to any line; the amount of two columns means value of other line. As a result, a strange variety periods a much amount is definitely an even number.
Exactly the same applies to the items of two unusual phone numbers.
Making a multiplication chart from storage
Building a multiplication chart from memory space can help youngsters learn the diverse numbers in the periods desks. This straightforward exercise enables your son or daughter to memorize the
amounts and discover how you can flourish them, which will help them in the future when they discover more challenging arithmetic. For any entertaining and good way to remember the phone numbers, you
may organize coloured switches in order that each corresponds to particular periods dinner table amount. Be sure to label every single row “1” and “” so that you can rapidly determine which variety
will come initial.
As soon as young children have learned the multiplication graph or chart bar from recollection, they ought to commit on their own to the task. This is why it is advisable to use a worksheet rather
than a conventional notebook computer to practice. Colourful and animated character templates can appeal to the detects of your respective children. Before they move on to the next step, let them
color every correct answer. Then, show the graph inside their study area or bed rooms to serve as a reminder.
By using a multiplication graph in everyday life
A multiplication graph shows you how to increase figures, anyone to 15. Additionally, it demonstrates the item of two figures. It may be beneficial in your everyday living, for example when splitting
up cash or gathering details on individuals. The subsequent are among the techniques you can use a multiplication chart. Use them to aid your kids comprehend the concept. We now have pointed out just
a few of the most common purposes of multiplication desks.
You can use a multiplication graph to help your youngster learn how to minimize fractions. The secret to success is to keep to the denominator and numerator on the left. Using this method, they will
likely see that a fraction like 4/6 could be decreased to a small part of 2/3. Multiplication maps are particularly ideal for young children since they help them to identify number patterns. You can
get Totally free computer versions of multiplication graph cafes on-line.
Gallery of Multiple Stacked Bar Chart Power Bi
Gr fico De L neas Y Columnas Apiladas En Power BI R Marketing Digital
Power BI Stacked Bar Chart Example Power BI Docs
Power BI Clustered Stacked Column Bar DEFTeam Power BI Chart
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Election 4 - math word problem (7582)
Election 4
In a certain election, there are three presidential candidates: 5 for secretory and 2 for treasurer. Find how many ways the election may (turn out/be held).
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
Showing 1 comment:
Help me know alot of mathematics questions
Tips for related online calculators
You need to know the following knowledge to solve this word math problem:
Grade of the word problem:
Related math problems and questions:
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C Program to swap first and last digit of a number
In this article, we will write a simple C program to swap the first and last digit of a number. For example, if the input number is 3456, then the program should print 6453 (swapping first and last
digit) as output.
C Program to swap the first and last digit of a number
In this program, we first counts the number of digits in the input number using while loop, then we determine the first and last digit using division and modulo operators. Later in this program, we
swap these digits using arithmetic operations.
The explanation of the program is at the end of this code.
#include <stdio.h>
int main() {
int num, temp, firstDigit, lastDigit, digitsCount = 0;
// Input number
printf("Enter a number: ");
scanf("%d", &num);
temp = num;
// Count number of digits
while (temp != 0) {
temp /= 10;
// Finds first and last digits of the input number
firstDigit = num / (int)pow(10, digitsCount - 1);
lastDigit = num % 10;
// Swap first and last digits of the number
temp = num - (firstDigit * (int)pow(10, digitsCount - 1));
temp += (lastDigit * (int)pow(10, digitsCount - 1));
temp -= lastDigit;
temp += firstDigit;
// Output the number with swapped digits
printf("Number with first and last digits swapped: %d\n", temp);
return 0;
Enter a number: 12345
Number with first and last digits swapped: 52341
In this example, user entered the number 12345. The first digit is 1, and the last digit is 5. The program swapped these first and last digits and printed the number 52341.
Explanation of the program:
1. Variable Declaration: In the main function, we declared the following variables:
□ num: This stores the number entered by the user.
□ temp: a temporary variable used for calculations.
□ firstDigit and lastDigit: to store the first and last digits of the input number.
□ digitsCount: to count the number of digits in the input number.
2. Input from user: We used printf() function to prompt the user for entering a number, the entered number is stored in the variable num using scanf() function.
3. Counting Digits: The program counts the number of digits in the input number using a while loop. It is done by dividing the input number by 10, this effectively removes a digit from the number,
at the same time we are increasing the value of digitsCount by 1. This process continues until the number becomes zero, at this moment the digitsCount contains the number of digits in the number.
4. Finding the First and Last Digits: The first and last digits are calculated based on the following logic:
□ firstDigit: This is calculated by dividing the input number by 10 raised to the power of (digitsCount - 1). For example, if I need to find the first digits in number 20, this would be 20/(10^
(2-1)) = 20/10 = 2, Similarly for 123 it would be 123/(10^(3-1)) = 123/100 = 1
□ lastDigit: This is equal to the remainder after dividing the number by 10, which can be easily done by modulo operator.
5. Swapping first and last Digits: The program swaps the first and last digits of the input number:
6. Output: Finally, the program prints the output number using printf() after swapping the first and last digits.
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Aernout van Enter
Groningen University (Netherlands)
One-sided versus two-sided regularity properties
Ergodic Theory and Dynamical Systems Seminar
17th March 2022, 2:00 pm – 3:00 pm
Fry Building, 2.04
Stochastic processes can be parametrised by time (such as occurs in Markov chains), in which case conditioning is one-sided (on the past), as naturally occurs in dynamical systems, or by
one-dimensional space (which is the case, for example, for one-dimensional Markov fields), as is natural in statistical mechanics, where the conditioning is two-sided (on the right and on the left).
I will discuss some examples, in particular generalising this distinction to g-measures versus Gibbs measures, where, instead of a Markovian dependence, the weaker property of continuity (in the
product topology) is considered. In particular I will discuss when the two descriptions (one-sided or two-sided) produce the same objects and when they are different. We show moreover the role
one-dimensional entropic repulsion plays in this setting. Based on joint work with R. Bissacot, E. Endo and A. Le Ny, and S. Shlosman.
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Finance for Executives
An Ideal Resource for Managers
• A focus on value creation improves your ability to make smart financial decisions.
• Each topic is approached from a problem-solving perspective. You will learn to solve the practical financial problems that executives face every day.
• Self-contained chapters make the text an ideal quick reference guide to finance. Review questions — with detailed answers — make it a valuable self-study tool.
• The distinguished author team shares their extensive experience teaching executives around the world.
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Periodic Reporting for period 1 - QuEST (Quantum Energy Conditions and Singularity Theorems) | H2020 | CORDIS | European Commission
Periodic Reporting for period 1 - QuEST (Quantum Energy Conditions and Singularity Theorems)
Periodo di rendicontazione: 2017-09-01 al 2019-08-31
In the context of general relativity, a singularity is defined as a boundary point of spacetime beyond which no extension is possible, and a singularity theorem is a proof that singularities are
inevitable under certain conditions. Although singularities were discovered early in the history of general relativity in solutions to Einstein’s field equations, the question of whether
singularities occur in the physical universe (e.g. inside black holes or at the big bang) remains open and continues to intrigue both mathematicians and physicists. Work on singularities can help
answer questions on the origin of our Universe and give insights into a complete theory that would unify gravity with other forces.
Fifty years ago, Hawking and Penrose developed the first general singularity theorems in classical General Relativity. These theorems showed that singularities exist in any spacetime that satisfies
certain properties. Some of these properties are mild assumptions on spacetime geometry but others depend on matter content and are more problematic. For example, Hawking assumed a Strong Energy
Condition (SEC) asserting positivity of the effective energy density (EED). Unfortunately quantised matter as described by quantum field theory (QFT) allows states that violate these energy
conditions and so it is necessary to try to prove singularity theorems under weaker assumptions. The main goal of the research project was to establish mathematically rigorous singularity theorems
for quantized matter fields based on quantum energy inequalities (QEIs) that constrain local averages of quantities like the EED.
Final period conclusions: During the project considerable progress was made towards deriving singularity theorems for quantised matter, and work is nearly complete on a Hawking-type singularity
theorem using hypotheses satisfied by a quantum field theory. Along the way, we have derived a classical singularity theorem for the Einstein-Klein-Gordon system, a QSEI for nonminimally coupled
scalar fields and a new method for proving singularity theorems (of both types) with weakened energy conditions. Additional results were derived during the duration of the project or are expected in
the near future and they are discussed below.
The project had four main outcomes.
1. We have proved a singularity theorem for the classical Einstein-Klein-Gordon theory in which the SEC can be violated, thus preventing the use of Hawking's original theorem. We first derived lower
bounds on local averages of the effective energy density (EED) for solutions to the Klein–Gordon equation, which were then used to prove a singularity theorem. This shows that all solutions of this
theory with sufficient initial contraction at a compact Cauchy surface will be future timelike geodesically incomplete. The required initial contraction was calculated for cosmological applications.
(Publication [1].)
2. We have derived a mathematically rigorous quantum strong energy inequality (QSEI) for nonminimally coupled scalar fields valid in general spacetimes. As had been anticipated, these QSEIs depend on
the state of interest. The state-dependence of these bounds in Minkowski spacetime for thermal (KMS) states was analyzed, and it was shown that the lower bounds grow more slowly in magnitude than the
EED itself as temperature increases. The lower bounds are therefore of lower energetic order than the EED, and qualify as nontrivial state-dependent QEIs. (Publication [2].)
3. We have developed a new method of proving singularity theorems with weakened energy conditions that avoids the Raychaudhuri equation but instead makes use of index form methods. These results
improve over existing methods and can be applied to hypotheses inspired by QEIs. In that case, quantitative estimates of the initial conditions required for our singularity theorems to apply were
made. (Reference [3]; currently under peer-review.)
4. Finally, we have made progress towards the first derivation of a semiclassical singularity theorem, combining the methods of part 3 with the QSEI bound of part 2. This joint work of the ER and the
supervisor is expected to appear as a pre-print in the near future.
A short summary of parts 1, 2 and 4 has been produced as a conference proceedings article [4] for the Proceedings of the 15th Marcel Grossman meeting.
[1] PJ Brown, CJ Fewster and E-A Kontou, A singularity theorem for Einstein-Klein-Gordon theory. General Relativity and Gravitation 50 (2018) 121 (24pp). DOI: 10.1007/s10714-018-2446-5.
[2] CJ Fewster and E-A Kontou, Quantum strong energy inequalities. Phys. Rev. D 99 (2019) 045001 (17pp). DOI: 10.1103/PhysRevD.99.045001 arXiv:1809.05047
[3] CJ Fewster and E-A Kontou, A new derivation of singularity theorems with weakened energy hypotheses (27pp). arXiv:1907.13604
[4] PJ Brown, CJ Fewster and E-A Kontou, Classical and quantum strong energy inequalities and the Hawking singularity theorem (6pp). arXiv:1904.00419
All the completed projects described in the previous section of the report represent progress beyond the state of the art. Specifically:
* Our index-form method is a simpler and much more general way to prove singularity theorems under weakened conditions than previous methods.
* No classical or quantum inequality bounds were previously known for theories violating the strong energy condition.
* When completed, our current project will be the first singularity theorem for a matter described by quantum fields under general conditions.
This is theoretical work without direct technological applications in view. However, it attracts significant public interest, reflected by the audiences at the two public outreach events during this
project. The first was an event “Travels in time, fiction and physics” at the Festival of Ideas in York, a popular festival featuring over 150 events every year including a wide range of public talks
from experts. The ER coordinated the event, which included presentations from the ER, the supervisor and two historians of science from the University of York as well as discussion with the audience.
The event was highly successful with over 160 attendees and positive feedback.
The second event was a public talk “Can we create wormholes using quantum fields?” by the ER as part of the “Pint of Science” series - a worldwide science festival which brings researchers to local
pubs to present their scientific discoveries. There are 600 events every year across the UK. The event sold out and received highly positive feedback.
During the fellowship the ER delivered over 15 talks in invited seminars as well as international and local conferences. For example, she spoke at the 15th Marcel Grossman meeting, GR22, APS meeting
and delivered seminars at the Perimeter Institute, Penn State University and the University of Nottingham. She also gave a talk at the North British Mathematical Physics Seminar.
As part of the fellowship, the two-day workshop “Energy conditions in quantum field theory and gravity” was organized at the University of York by the ER and the supervisor. The workshop was
considered very successful with a total of 35 participants and 10 invited speakers, all of whom are international experts in the field.
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price to cash flow ratio calculator
Price To Cash Flow Ratio Calculator
Calculating the price-to-cash-flow (P/CF) ratio is crucial for investors to assess a company’s financial health and valuation. This ratio compares a company’s market price per share to its operating
cash flow per share. In this article, we’ll provide a handy calculator to simplify this calculation process.
How to Use
Enter the required values in the respective fields of the calculator below and click on the “Calculate” button to obtain the price-to-cash-flow ratio.
The formula for calculating the price-to-cash-flow (P/CF) ratio is:
Example Solve
Let’s consider a hypothetical company with a market price per share of $50 and an operating cash flow per share of $5.
Q: What does the price-to-cash-flow ratio indicate?
A: The price-to-cash-flow ratio provides insights into how much investors are willing to pay for a company’s cash flow.
Q: How does the P/CF ratio differ from other valuation metrics?
A: While the price-to-earnings (P/E) ratio focuses on earnings, the P/CF ratio emphasizes cash flow, which can offer a clearer picture of a company’s financial health.
Q: Is a higher or lower P/CF ratio preferable?
A: A lower P/CF ratio suggests that the company may be undervalued, while a higher ratio could indicate overvaluation.
The price-to-cash-flow (P/CF) ratio is a valuable tool for investors to assess a company’s valuation and financial performance. By utilizing the calculator provided below, investors can make informed
decisions regarding their investments.
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String Concatenation in R | R-bloggersString Concatenation in R
String Concatenation in R
[This article was first published on
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
String concatenation is a rather basic function – but my particular programming reflexes did not help me figure out how to do this in R. I tried the + and & operator, and even the || operator to no
avail. Also tried concat() function… no dice. The answer?
> paste(‘this string is concatenated’, ‘to this string’)
Not exactly intuitive eh? Even more confusing since R uses a lot of UNIX idioms… and in this case, the paste command (which in UNIX is used to combine files) performs a rather different role. A
couple of itmes to note:
• The two strings above are concatenated with a space between them!
• Additional strings can be included as arguments:
• If you use a vector for an argument, you can use the collapse parameter to achieve the same sort of result:
> paste(c(‘a’,’b’,’c’), collapse=’ ‘)
• As always, more information is available through R’s help system:
> ?paste
I found this exercise fascinating – so many things in R just “work” like I hope they will. Many functions take lists, vectors, or whatever is thrown their way and do the “right” thing with them.
There is plenty of C like syntax that is familiar – but there are significant differences that just need to be memorized. String concatenation seems to fit into this category.
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not rational is called irrational Irrational numbers include pi phi square roots etc
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The prom committee has decided to create a replica of the St. Louis arch as a gateway to enter the prom. Because of the height of the ceiling they have decided that the arch should be 8 feet high.
They don't want the arch to be too wide or too narrow. After much discussion they finally agreed that at a height of 6 feet the arch must be 4 feet wide.
Your job is to build the arch. Research the arch on the internet to see what it looks like and decide what materials would be good to use to make it look realistic.
Use the links below to help you determine an equation for the arch. Use the equation to determine the width of the arch at its base.
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KNN | Jason Siu
Machine Learning
Machine Learning
1. Partition Clustering (Non-Hierarchical)
☆ Divide the data objects into unique subsets, of which a number of sets (i.e., ) are pre-defined.
☆ are sequential either building up from separate clusters or breaking down from the one cluster.
☆ In some cases, the exact number of clusters may be known (like marketing customers group), then we can use Partition Clustering.
☆ A good clustering is one for which the within-cluster variation is as small as possible.
Knowing what Partition Clustering is, let's explore one of its kind — KNN.
• Each cluster is associated with a centroid
• Each point is assigned to the cluster with the CLOSEST centroid.
1. The data is in feature space, which means data in feature space can be measured by distance metrics such as Manhattan, Euclidean etc.
2. Each data point can only belong to one cluster.
3. Each training data point consists of a set of vectors and class labels associated with each vector.
4. Wishes to have ‘K’ as an odd number in case of 2 class classification.
Mechanism - Here are the steps of KNN :
1. Select k points (at random) as the initial centroids
2. Repeat :
2.1. Form k clusters by assigning all points to the CLOSEST centroid
2.2. Re-compute the centroid of each cluster
3. Until the centroids don’t change.
Why and Why not KNN :
Advantage :
• Since the KNN algorithm requires no training before making predictions, new data can be added seamlessly which will not impact the algorithm's accuracy.
Disadvantage :
It depends on initial values. The problem of which is that it has different :
R code
set.seed(2021) # set seed for reproducibility (kmeans have initialise at different random starts)
flea_km <- stats::kmeans(flea_std[,2:7],
centers = 3) ## flea_std is a numeric df.
Elbow plot
# factoextra::fviz_nbclust helper function; used; visualise & determine the optimal k
factoextra::fviz_nbclust(x = tourr::flea %>% select(-species),
FUNcluster = kmeans,
method = "wss", # within cluster ss
k.max = 10) # max no. of clusters to consider
R interpretation
Comparing HCluster
assumed you have run this
# --- (II) (III) compute distance matrix + pass as input into `stats::hclust`
flea_hc_w <- stats::dist(flea_std[,2:7]) %>%
stats::hclust(method = "ward.D2")
map cluster labels to obs. from both methods
flea_std <- flea_std %>%
mutate(cl_w = stats::cutree(flea_hc_w, k = 3), # extract cluster solution from hclust; k = 3
cl_km = flea_km$cluster) # extract cluster solutions from k-means; k = 3
compute confusion table; compare both results
•note: labels doesn't matter
flea_std %>%
count(cl_w, cl_km) %>% # compute agreemeent
# pivot into wide form; create confusion table
pivot_wider(names_from = cl_w,
values_from = n,
values_fill = 0) %>%
rename("cl_km/cl_wards" = cl_km)
b. Map the cluster labels from the two results, and calculate the agreement.
## matrix
table(actual = flea_std$species, fitted = flea_km$cluster)
KNN and SSE
• Split ‘loose’ clusters, i.e., clusters with relatively HIGH SSE.
□ If you have 100 data points, and you have 1 cluster, then your SSE is highest.
• Merge clusters that are ‘close’ and that have relatively LOW SSE.
□ If you have 100 data points, and you have 100 clusters, then your SSE is 0.
What are the advantages and disadvantages of hierarchical & non-hierarchical clustering
Hierarchical: are sequential either building up from separate clusters or breaking down from the one cluster.
Non-hierarchical: The number of clusters is chosen a priori.
Advantages of Hierarchical clustering
• All possible solutions (with respect to the number of clusters) is provided in a single analysis
• It is structured and can be explored using the dendrogram.
• Don’t need to assume any particular number of clusters. Any desired number of clusters can be obtained by ‘cutting’ the dendrogram at the proper level
Advantages of K-Means Clustering
• Can explore cluster allocations that will never be visited by the path of hierarchical solutions.
• With a large number of variables, if K is small, K-Means may be computationally faster than hierarchical clustering.
• K-Means might yield tighter clusters than hierarchical clustering
• An instance can change cluster (move to another cluster) when the centroids are recomputed.
Disadvantages of Hierarchical clustering
• It is not possible to undo the previous step: once the instances have been assigned to a cluster, they can no longer be moved around.
• Time complexity: not suitable for large datasets
• Very sensitive to outliers
Disadvantages of Hierarchical clustering
• Need to assume any particular number of clusters (K-Value), which is hard to predict.
• Initial seeds (centers of each cluster) have a strong influence on the final results
How does each point find its closest centroid?
• To find its centroid, you need to calculate the "distance" between each point and all the centroids,
□ then you will know which is the closest one.
Is there any Pre-processing for KNN?
Yes, you need to :
• Normalise the data • Eliminate outliers
Is there any Post-processing for KNN?
• Eliminate small clusters that may represent outliers 埋吾到堆ge 走開
－Split ‘loose’ clusters, i.e., clusters with relatively high SSE.
－Merge clusters that are ‘close’ and that have relatively low SSE
Will the initial centroids influence final clusters? if so, how do you cope with that?Why is KNN tending to produce a good clustering?
Because the within-cluster variation tends to be small.
By partitioning K clusters, we want to minimise the total within-cluster variation as SMALL as possible.
Extra Resource
• Author:Jason Siu
• Copyright:All articles in this blog, except for special statements, adopt BY-NC-SA agreement. Please indicate the source!
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Four from MIT awarded 2021 New Horizons in Physics and New Frontiers in Mathematics prizesFour from MIT awarded 2021 New Horizons in Physics and New Frontiers in Mathematics prizes » MIT Physics
Images: courtesy of the winners
Images: courtesy of the winners
Four from MIT awarded 2021 New Horizons in Physics and New Frontiers in Mathematics prizes
Physicists Tracy Slatyer and Netta Engelhardt and mathematicians Lisa Piccirillo and Nina Holden PhD ’18 are honored by the Breakthrough Prize Foundation.
Three MIT faculty members and one alumna have been named winners of prizes awarded by the Breakthrough Prize Foundation, which honor early-career achievements in the fields of physics and
Physicists Tracy Slatyer and Netta Engelhardt will each receive the 2021 New Horizons in Physics Prize, an award of $100,000 that recognizes promising junior researchers who have produced important
work in their field.
Mathematicians Lisa Piccirillo and Nina Holden PhD ’18 will each receive the 2021 Maryam Mirzakhani New Frontiers Prize, a $50,000 award that recognizes outstanding early-career women in mathematics.
The prize was created in 2019 in honor of Iranian mathematician and Fields Medalist Maryam Mirzakhani, who made groundbreaking contributions to her field before her death in 2017 at the age of 40
after battling breast cancer.
“The New Horizons and New Frontiers prizes recognize some of the most talented young researchers in the world,” says Nergis Mavalvala, Dean of MIT’s School of Science, and the Curtis and Kathleen
Marble Professor of Astrophysics. “I’m thrilled that four colleagues from MIT have been recognized as amazing researchers who have already done important and impactful work. I can’t wait to see what
comes next from these brilliant young women.”
Both prizes are part of a family of awards given out each year by the Breakthrough Prize Foundation and its founding sponsors — Sergey Brin, Priscilla Chan and Mark Zuckerberg, Ma Huateng, Yuri and
Julia Milner, and Anne Wojcicki. The winners are chosen by a committee of past awardees in each field.
Dark matter patterns
Tracy Slatyer is the Jerrold R. Zacharias Career Development Associate Professor of Physics at MIT, and will receive the 2021 New Horizons in Physics Prize “for major contributions to particle
astrophysics, from models of dark matter to the discovery of the ‘Fermi Bubbles,’” according to the award citation.
A theoretical physicist who works on particle physics, cosmology, and astrophysics, Slatyer has pioneered new techniques to search through telescope data for clues to the nature and interactions of
dark matter, which is thought to make up more than 80 percent of the matter in the universe. She has used data from the Fermi Gamma-Ray Space Telescope to co-discover the Fermi bubbles, a mysterious
structure of high-energy gamma rays bubbling out from the center of the Milky Way galaxy.
Slatyer grew up in Canberra, Australia, and received her undergraduate degree from the Australian National University. She carried out postgraduate research at the University of Melbourne before
moving to Boston in 2006 as a graduate student at Harvard University. She worked as a postdoc at the Institute for Advanced Study in Princeton, New Jersey, for three years before joining the MIT
faculty as a member of the Center for Theoretical Physics in 2013. She is the recipient of the Henry Primakoff Award for Early-Career Particle Physics and a Presidential Early Career Award for
Scientists and Engineers.
Black hole information
Netta Engelhardt is an assistant professor of physics and a member of the Center for Theoretical Physics at MIT. She shares the 2021 New Horizons in Physics Prize with three others: Ahmed Almheiri of
the Institute for Advanced Study, Henry Maxfield of the University of California at Santa Barbara, and Geoff Penington of Stanford University. The prize recognizes the researchers “for calculating
the quantum information content of a black hole and its radiation.”
Black holes are thought to contain a huge amount of information in the form of the matter that falls into them. They are also known to emit radiation in response to quantum jiggling of their
surroundings. To what extent information is released along with this radiation has been a conflicting question for both quantum mechanics and Einstein’s theory of gravity.
Engelhardt, Almheiri, and Maxfield found that as matter falls into the black hole, the information that it contains increases. As the black hole ages, it gives off radiation and in the process spews
information back out. Penington came to the same conclusion independently. Together, the researchers’ work showed that information can indeed safely escape a black hole.
Engelhardt’s research focuses on gravitational aspects of quantum gravity, and on understanding the predictions of quantum gravity in the context of gravitational singularities — locations in
space-time where the gravitational field of an astrophysical object is predicted to be infinite. She is investigating the black hole information paradox, thermodynamic behavior of black holes, and
the idea that singularities are always hidden behind event horizons.
Engelhardt grew up in Jerusalem and Boston. She received her undergraduate degree from Brandeis University and her PhD from the University of California at Santa Barbara. She was a postdoc at
Princeton University and a member of the Princeton Gravity Initiative before joining the MIT physics faculty in July 2019.
A tangled proof
Lisa Piccirillo is an assistant professor of mathematics at MIT. She is being awarded the 2021 Maryam Mirzakhani New Frontiers Prize “for resolving the classic problem that the Conway knot is not
smoothly slice,” according to the award citation.
For decades, the Conway knot was an unsolved problem in the subfield of mathematics known as knot theory. One of the fundamental questions that knot theorists try to puzzle out is whether a knot is a
“slice” of a more complicated, higher-order knot. Mathematicians have determined the “sliceness” of thousands of knots with 12 or fewer crossings, except for one: the Conway knot. Named after the
mathematician John Horton Conway, this knot consists of 11 crossings, the sliceness of which mathematicians had struggled for decades to explain.
Piccirillo heard about the Conway knot problem as a graduate student at the University of Texas at Austin. Over one short week in the summer of 2018, she solved the puzzle, with a proof that showed
the tricky knot was in fact not a slice of a higher-order knot. Her work, and the classical tools she used to lay out her proof, were published earlier this year — a feat that generated widespread
interest beyond the mathematics community.
Piccirillo was raised in Greenwood, Maine, and earned an undergraduate degree from Boston College. She earned a PhD in low-dimensional topology at the University of Texas at Austin and worked as a
postdoc at Brandeis University before joining the MIT faculty in July 2020.
“Random triangulations”
Nina Holden PhD ’18 is a junior fellow at the Institute for Theoretical Studies at ETH Zurich, and an alumna who earned a doctorate in mathematics from MIT in 2018. She is a recipient of the 2021
Maryam Mirzakhani New Frontiers Prize “for work in random geometry, particularly on Liouville Quantum Gravity as a scaling limit of random triangulations.”
Holden’s research focuses on probability theory, and in particular, conformally invariant probability. She studies universal models for random surfaces, which are of interest in mathematical physics
and theoretical probability.
A classical result in probability theory is that a random walk converges to the continuum random process known as Brownian motion when the lattice size is sent to zero. Holden has proved similar
convergence results for discrete random surfaces known as random planar maps. In particular, she proved in joint work with Xin Sun that certain random planar maps known as triangulations converge to
the continuum random surfaces known as Liouville quantum gravity surfaces. This confirms predictions made in string theory and conformal field theory in the 1980s, and is also of fundamental interest
in theoretical probability theory.
Holden earned her bachelor’s and master’s degrees in mathematics from the University of Oslo before pursuing a PhD in mathematics at MIT, where she studied with Scott Sheffield, the Leighton Family
Professor of Mathematics. She will be an associate professor at the Courant Institute of Mathematical Sciences at New York University in 2021.
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Threshold coupling strength for equilibration between small systems
In this paper we study the thermal equilibration of small bipartite Bose-Hubbard systems, both quantum mechanically and in mean-field approximation. In particular we consider small systems composed
of a single-mode "thermometer" coupled to a three-mode "bath," with no additional environment acting on the four-mode system, and test the hypothesis that the thermometer will thermalize if and only
if the bath is chaotic. We find that chaos in the bath alone is neither necessary nor sufficient for equilibration in these isolated four-mode systems. The two subsystems can thermalize if the
combined system is chaotic even when neither subsystem is chaotic in isolation, and under full quantum dynamics there is a minimum coupling strength between the thermometer and the bath below which
the system does not thermalize even if the bath itself is chaotic. We show that the quantum coupling threshold scales like 1 /N (where N is the total particle number), so that the classical results
are obtained in the limit N →∞ .
Physical Review A
Pub Date:
June 2019
Phys. Rev. A 99, 063617 (2019)
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How to Find the Median: A Comprehensive Guide
Introduction: Understanding the Basics
Greetings readers! Are you struggling with finding the median of a set of numbers? Do you find this topic confusing? No worries, this article will guide you through the process of finding the median
in a simple and easy-to-understand manner. Before we dive into the details of how to find the median, let’s discuss what it is and why it’s important.
The median is a statistical measure that represents the central value of a set of data, separating it into two equal parts. It is an important tool for analyzing data as it helps you understand the
distribution of values within a dataset. By finding the median, you can determine the mid-point of the data and understand how it is distributed around this value. This is useful in various fields,
including finance, economics, and social sciences, among others.
Now that we have a basic idea of what the median is and its importance, let’s move on to the main topic of this article – how to find the median.
Finding the Median: Step-by-Step Guide
Step 1: Arrange the data in ascending or descending order.
The first step in finding the median is arranging the dataset in either ascending or descending order. This makes it easier to identify the value that represents the midpoint of the dataset. For
instance, if we have the following dataset:
We would arrange it in ascending order as:
Step 2: Determine the number of values in the dataset.
After arranging the data in order, the next step is to count the number of values in the dataset. This will help you determine whether the dataset has an even or odd number of values.
Step 3: Identify the middle value.
If the dataset has an odd number of values, the median is the middle value. For example, in the dataset above, we have five values, so the middle value is 15. Therefore, the median is 15.
If the dataset has an even number of values, the median is the average of the two middle values. To illustrate this, let’s consider the following dataset:
Since this dataset has six values, we need to find the average of the two middle values, which are 15 and 20. To do this, we add the two values and divide the sum by two as follows:
(15 + 20)/2 = 17.5
Therefore, the median of this dataset is 17.5.
Step 4: Interpret the results.
Once you have found the median, it’s important to interpret the results in the context of the dataset. The median is a measure of central tendency and can provide insight into the distribution of
values within the dataset. For example, if the median is significantly different from the mean, it may indicate that the data is skewed or has outliers.
Frequently Asked Questions (FAQs)
Q1: What is the difference between the median and the mean?
The mean and median are both measures of central tendency but are calculated differently. The mean is the arithmetic average of all the values in a dataset, while the median is the middle value when
the dataset is arranged in either ascending or descending order.
Q2: What is the advantage of using the median over the mean?
The median is less sensitive to outliers in the data compared to the mean. This means that if there are extreme values in the dataset, the median will be less affected than the mean.
Q3: Can the median be used for categorical data?
No, the median can only be used for numerical data. For categorical data, other measures such as mode or frequency distribution are used.
Q4: What is the median used for in finance?
The median is often used in finance to calculate the mid-point of a range of values or to determine the average income or expenditure of a group.
Q5: Is the median the same as the mode?
No, the median and mode are different measures of central tendency. The mode is the value that occurs most frequently in a dataset, while the median is the middle value.
Q6: How do you calculate the median in Excel?
To calculate the median in Excel, use the MEDIAN function. For example, to find the median of a range of cells, enter the following formula into a blank cell: =MEDIAN(range). Replace “range” with the
actual range of cells containing the data.
Q7: Can the median value change if additional values are added to the dataset?
Yes, the median value can change if additional values are added or if existing values are removed from the dataset.
Q8: Does the median always exist in a dataset?
Yes, the median always exists in a dataset, even if it is not a whole number. For example, the median of the dataset {1, 2, 3.5, 7} is 2.75.
Q9: How do you find the median of a set of decimals?
To find the median of a set of decimals, arrange the data in order and determine the middle value. If the dataset has an even number of values, calculate the average of the two middle values.
Q10: Can the median be used for skewed data?
Yes, the median can be used for skewed data. It is a robust measure of central tendency that is less sensitive to extreme values compared to the mean.
Q11: How do you find the median of a grouped frequency distribution?
To find the median of a grouped frequency distribution, first calculate the cumulative frequency for each interval. Then, find the interval that contains the middle value and calculate the median
using the formula:
Median = L + [(n/2 – CF)/f] x i
L = lower class limit of the interval containing the median
n = total number of values in the dataset
CF = cumulative frequency of the interval immediately preceding the median interval
f = frequency of the median interval
i = interval width
Q12: Can the median be higher than the highest value in the dataset?
No, the median cannot be higher than the highest value in the dataset because it is a value within the dataset. If the median is higher than the highest value, it indicates an error in calculating
the median.
Q13: How do you find the median of a dataset with missing values?
If a dataset has missing values, you can still find the median by excluding the missing values from the calculation. If there are too many missing values, it may be necessary to use alternative
measures of central tendency.
Conclusion: Taking Action
In conclusion, finding the median is a simple yet essential statistical tool that can help you understand the distribution of values within a dataset. By following the steps outlined in this article,
you can easily find the median of any dataset. Remember, the median is just one of many measures of central tendency, and it’s important to interpret it in the context of the data you’re working
We hope this guide has been helpful in understanding how to find the median. If you have any further questions or need clarification on any of the steps, don’t hesitate to leave a comment below!
Take action now!
Practice finding the median using different datasets and see how it can provide insight into the distribution of values. Share this article with your friends and colleagues who may find it helpful in
their work.
Closing Disclaimer
This article is provided for educational and informational purposes only. We do not guarantee its accuracy or completeness. The information provided in this article is not intended to be a substitute
for professional advice. Always seek the advice of a qualified professional with any questions you may have regarding a specific issue.
Cuplikan video:How to Find the Median: A Comprehensive Guide
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List of Participants, Organizing and Advisory Committees
A. Zaitsev (IHEP, Protvino)
L. Soloviev ( IHEP, Protvino )
Y.-S. Kim ( Maryland University)
F. Tikhonin ( IHEP, Protvino)
H. Terazawa ( KEK, Tokyo)
A. Smirnov ( Yaroslavl University)
A. Pechenkin ( IHST RAS, Moscow)
V. Khruschev ( VNIIMS, Moscow)
O. Pavlovsky ( Bogolyubov ITPM, Moscow)
C. Marchal (ONERA, Chatillon)
A. Logunov, M. Mestvirishvili (IHEP, Protvino)
A. Genk (St.-Petersburg Association of Scientists)
A. Tyapkin (JINR, Dubna )
A. Blanovsky (Teacher Tech. Center, Los-Angeles)
I. Schmelzer (WIAS, Berlin)
A. Zakharov (ITEP, Moscow)
A. Burinskii (Nuclear Safety Inst. of RAS, Moscow)
Spinning Particle as Super-Kerr Black Hole with Broken N=2 Supersymmetry
(Paper was not received)
Yu. Vyblyi (Inst. of Physics, Minsk)
The Nonsymmetrical Tensor Potential in the Relativistic Theory of Gravitation
(Paper was not received)
D. Bayuk (IHST RAS, Moscow)
G. Harigel (CERN, Geneva)
The Legacy of Radioactive Waste from the Weapons Laboratories: How to Render the Radioactive Waste Harmless?
O. Khrustalev (Bogolyubov ITPM, Moscow)
Quantum Computing
(Paper was not received)
B. Arbuzov (IHEP, Protvino)
M. Tchitchikina (Lomonosov MSU, Moscow)
S. Vernov (Skobeltsyn INR, Moscow)
N. Lunin (IAP RAS, Nizhni Novgorod)
K. Tomilin (IHST RAS, Moscow)
V. Petrov (IHEP, Protvino)
All conferences 99 Support team
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Regression Analysis Microsoft Excel 9780789756558
DOC EP2100 tenta 180428 FINAL FINAL LSG Siw Zix
It explores main concepts from basic to expert level which can help you achieve better grades, develop your academic career, apply your knowledge at work or do your business forecasting research.
Multiple Regression in Excel in a nutshell. Focusing on Excel functionality more than presentation of regression theory. Multiple regressions to excel.
Most or all P-values should be below below 0.05. In our example this is the case. (0.000, 0.001 and 0.005). Coefficients. The regression line is: y = Quantity Sold = 8536.214-835.722 * Price + 0.592
* Advertising. In other words, for each unit increase in price, Quantity Sold decreases with 835.722 units.
Linjär regression - Miljostatistik.se
The cross-validation results were imported to Microsoft Excel for further processing, and the. consultancies is the application of multivariate statistics to provide actionable in a call centre
working environment using multiple regression analyses. regressionsanalyser med hjälp av Excel och statistikprogrammet IBM SPSS. been analyzed in multiple regression using Excel and the IBM SPSS
statistics Multi-level regression model on multiply imputed data set in R (Amelia, zelig, lme4) Jag försöker skapa en VBA-kod i Excel för att köra integrerad funktion med Linear Layout interview
question screens candidates for knowledge of Android.
Regression Analysis Microsoft Excel 9780789756558
The following options appear on the four Multiple Linear Regression dialogs. Variables Regression Model Building with MS Excel: Using Excel's Multiple Regression.
Download the Mount Pleasant Real Estate. Data from stat.hawkeslearning.com and open it with Microsoft Excel.
Taxishare login
Viewed 760 times 3. 1. I want to export results from multiple regressions into an excel file in a very specific format . MWS. data 2020-12-24 Setting up a multiple linear regression After opening
XLSTAT, select the XLSTAT / Modeling data / Regression function. Once you've clicked on the button, the Linear Regression dialog box appears.
You may also look at these useful functions in excel – Formula of Coefficient of Determination; Non-Linear Regression in Excel; Regression vs. ANOVA; Formula of Multiple Learn multiple regression
analysis through a practical course with Microsoft Excel® using stocks, rates, prices and macroeconomic historical data. It explores main concepts from basic to expert level which can help you
achieve better grades, develop your academic career, apply your knowledge at work or do your business forecasting research. » Multiple Regression Analysis. Multiple Regression Analysis When to Use
Multiple Regression Analysis.
Om smarta ett fysiologiskt perspektiv
Excel Multiple Regression DEFINITION : Multiple regression is a method used in statistics to predict the outcome of a response or dependent variable using two or more explanatory or independent
variables. The Multiple Regression analysis gives us one plot for each independent variable versus the residuals. We can use these plots to evaluate if our sample data fit the variance’s assumptions
for Multiple Regression. In this part of the website, we extend the concepts from Linear Regressionto models that use more than one independent variable. We explore how to find the coefficients for
these multiple linear regression models using the method of least squares, how to determine whether independent variables are making a significant contribution to the model and the impact of
interactions between variables on the model. This article shows how to use Excel to perform multiple regression analysis.
Excel ist eine tolle Möglichkeit zum Ausführen multipler Regressionen, wenn ein Benutzer keinen Zugriff auf erweiterte Statistik-Software hat. Das Ganze geht schnell und lässt sich leicht erlernen.
The last method for regression is not so commonly used and requires statistical functions like slope (), intercept (), correl (), etc. to carry out regression analysis. Things to Remember About
Linear Regression in Excel. Regression analysis is generally used to see if there is a statistically significant relationship between two sets of variables. 2020-09-05 · The best method to do a
detailed regression analysis in Excel is to use the “Regression” tool which comes with Microsoft Excel Analysis ToolPak.
Forsta mobilen i varlden
Regression Analysis Microsoft Excel 9780789756558
Multiple Regression Analysis in Excel. Regression analysis describes the relationships between a set of independent variables and the dependent variable. It produces an equation where the
coefficients represent the relationship between each independent variable and the dependent variable. You can also use the equation to make predictions. Excel Functions: The functions SLOPE,
INTERCEPT, STEYX and FORECAST don’t work for multiple regression, but the functions TREND and LINEST do support multiple regression as does the Regression data analysis tool.
Leasingkontrakt vw
Linear Layout Android Adaface
ANOVA; Formula of Multiple 2017-02-02 2019-06-26 Example 3 - Multiple Linear Regression. Copy the example data in the following table, and paste it in cell A1 of a new Excel worksheet. For formulas
to show results, select them, press F2, and then press Enter. If you need to, you can adjust the column widths to see all the data. Excel: Build a Model to Predict Sales Based on Multiple Regression.
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Technology and Math | Pedagogy Non Grata
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Technology For Math Instruction
In this article I will review the findings for the 2022, meta-analysis on technology and math instruction, by Ran, Et al. This meta-analysis examined 77 studies on the topic. All studies had a
control group, reported sufficient statistical outcomes, were written in English, and included students ranging from Kindergarten to Grade 12. The meta-analysis excluded all studies from before the
year 2000, to ensure relevance to today's technology. The authors also did a sub-analysis on the dates of studies, comparing the effect sizes of older studies to newer ones. There were no meaningful
differences within this sub-analysis.
Meta-Analysis Interventions Glossary:
Collaboration Interventions:
Technology used to improve collaboration and communication for students showed the largest effects. Interventions that fit into this category did things like provide students virtual classrooms,
interaction opportunities between the teacher and classmates, or interventions that greatly “extend students' learning opportunities beyond the physical classrooms.” These results might lend credence
to the argument of virtual classrooms like Google Classroom, being used not as a replacement for the physical classroom, but as an addendum to.
Problem Solving Interventions:
Problem solving interventions were taught using technology, by providing students with problem solving questions on a computer, that also included visuals to enhance the students conceptual
understanding of the problem.
Conceptual Understanding Interventions:
Software programs that were designed to enhance students' conceptual knowledge. The authors specifically cited geometry programs such as GeoGebra as having the largest effect.
Adaptive Processes Interventions:
Interventions, in which the software automatically adjusted the difficulty and type of problems to students’ needs.
Formative Assessment:
Interventions, in which the technology was used to monitor student learning, but not matched with any specific follow up instruction.
The overall impact of technology interventions for math instruction was very low; indeed, the mean effect size found in this meta-analysis was barely statistically significant. However, this is
consistent with other meta-analyses on the topic. John Hattie’s meta-analysis of the topic, which included 911 total studies, found similar results, with a mean ES of .35. What likely matters is not
whether or not technology is used, but rather how it is used. Moreover, if we look at this study's sub analysis of intervention duration, we find that the longer an experiment continued the less
effective technology was. This possibly suggests that the primary usefulness of technology in the classroom, lies only in its novelty effect. One finding of this experiment that surprised me was the
futility in using technology for adaptive processes, as adaptive processes by definition seek to do what all good teachers should strive to do, individualize the instruction. One flaw in the
programmining studied, might be that while the software adapted questions, it may not have provided additional explicit instruction, where the students needed it. Overall, within this meta-analysis,
we see weak results for using technology to assist with math instruction, with the only meaningful outcomes found for collaborative technology interventions.
Ran. (2022). A meta‐analysis on the effects of technology’s functions and roles on students’ mathematics achievement in K‐12 classrooms. Journal of Computer Assisted Learning., 38(1), 258–284.
J, Hattie. (2022). Meta-X. Visible Learning. Retrieved from <https://www.visiblelearningmetax.com/influences>.
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Euclidean algorithm
From Encyclopedia of Mathematics
A method for finding the greatest common divisor of two integers, two polynomials (and, in general, two elements of a Euclidean ring) or the common measure of two intervals. It was described in
geometrical form in Euclid's Elements (3rd century B.C.).
For two positive integers
where the
The least positive remainder
The Euclidean algorithms for polynomials or for intervals are similar to the one for integers. In the case of incommensurable intervals the Euclidean algorithm leads to an infinite process.
The Euclidean algorithm to determine the greatest common divisor of two integers
A slight extension of the algorithm also yields a solution of
[a1] W.J. Leveque, "Topics in number theory" , 1 , Addison-Wesley (1956) pp. Chapt. 2
How to Cite This Entry:
Euclidean algorithm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euclidean_algorithm&oldid=16080
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
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S&P500 Sharpe ratio
In this exercise, you're going to calculate the Sharpe ratio of the S&P500, starting with pricing data only. In the next exercise, you'll do the same for the portfolio data, such that you can compare
the Sharpe ratios of the two.
Available for you is the price data from the S&P500 under sp500_value. The risk-free rate is available under rfr, which is conveniently set to zero. Let's give it a try!
This is a part of the course
“Introduction to Portfolio Analysis in Python”
View Course
Exercise instructions
• Calculate the total return of the S&P500 pricing data sp500_value using indexing and annualize the total return number; the data spans 4 years.
• Calculate the daily returns from the S&P500 pricing data, you'll need this for the volatility calculation.
• Calculate the standard deviation from the returns data and annualize the number using 250 trading days.
• Finally, calculate the Sharpe ratio using the annualized return and the annualized volatility and print the results.
Hands-on interactive exercise
Have a go at this exercise by completing this sample code.
# Calculate total return and annualized return from price data
total_return = (sp500_value[____] - ____[____]) / ____[____]
# Annualize the total return over 4 year
annualized_return = ((____ + ____)**(____/____))-1
# Create the returns data
returns_sp500 = ____.____()
# Calculate annualized volatility from the standard deviation
vol_sp500 = ____.____() * np.sqrt(____)
# Calculate the Sharpe ratio
sharpe_ratio = ((____ - rfr) / ____)
print (sharpe_ratio)
This exercise is part of the course
Introduction to Portfolio Analysis in Python
Learn how to calculate meaningful measures of risk and performance, and how to compile an optimal portfolio for the desired risk and return trade-off.
What is DataCamp?
Learn the data skills you need online at your own pace—from non-coding essentials to data science and machine learning.
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Steady state heat equation in a rectangle with a punkt heat source
• Thread starter fluidistic
• Start date
In summary, the person has been trying to solve a heat equation in a rectangle but has not been successful. They have tried using separation of variables and Green's function, but have not found a
solution. They then considered simplifying the problem by defining a new function, but that also did not work. When using separation of variables, they obtained solutions of the form X(x)Y(y), but
were unable to determine the constants. The person is seeking assistance with this problem.
Homework Statement
Solve the steady state heat equation in a rectangle whose bottom surface is kept at a fixed temperature, left and right sides are insulated and top side too, except for a point in a corner where
heat is generated constantly through time.
Relevant Equations
##\kappa \nabla ^2 T + g =0##
I have checked several textbooks about the heat equation in a rectangle and I have found none that deals with my exact problem. I have though to use separation of variables first (to no avail), then
Green's function (to no avail), then simplifying the problem for example by defining a new function in terms of ##T(x,y)## such that it would satisfy a homogeneous problem instead, but to no avail.
(is a problem even called homogeneous when ##dT/dx|_{x_0} = 0## rather than ##T(x=x_0)=0##? I guess not.)
Out of memory, when I went with separation of variables to tackle ##\kappa \left( \frac{\partial ^2 T}{\partial x^2}+ \frac{\partial ^2 T}{\partial y^2}\right) = 0##, I obtained solutions of the form
##X(x)Y(y)## with ##X(x)=A\cosh(\alpha x)+B\sinh(\alpha x)## and ##Y(y)=C\cos(\alpha y)+D\sin(\alpha y)## where ##\alpha## is the separation constant. The boundary conditions are of the type
Dirichlet for the bottom surface: ##T(x,y=0)=T_0##. And Neumann elsewhere: ##\frac{\partial T}{\partial x}|_{x=0, y=0}## for ##y\in [0,b)##, ##\frac{\partial T}{\partial x}|_{x=a, y=0}## for ##y\in
[0,b]## and ##\frac{\partial T}{\partial y}|_{x, y=b}## for ##x\in (0,a]##. The power generated translates as the Neumann boundary condition ##\nabla T \cdot \hat n## and so ##\frac{\partial T}{\
partial x}|_{x=0, y=b}+ \frac{\partial T}{\partial y}|_{x=0, y=b}=p## where ##p## is the power density of the heat source.
I have been stuck there, I could not get to apply and know the constants ##A##, ##B##, ##C## and ##D##, nor ##\alpha##. All of these constants are in fact depending on ##n##, natural numbers, because
the separable solutions are eigenfunctions, etc.
Any pointer would be appreciated. Thank you!
Staff Emeritus
Science Advisor
Homework Helper
2023 Award
I’m thinking of an array of sources and sinks.
FAQ: Steady state heat equation in a rectangle with a punkt heat source
1. What is the steady state heat equation?
The steady state heat equation is a mathematical representation of the flow of heat in a system that has reached a stable temperature. It takes into account factors such as heat sources, thermal
conductivity, and boundary conditions to determine the temperature distribution in the system.
2. How does the steady state heat equation apply to a rectangle?
In the context of heat transfer, a rectangle is a common shape used to represent a two-dimensional system. The steady state heat equation can be applied to a rectangle to calculate the temperature
distribution within the system, taking into account the dimensions of the rectangle and any heat sources or boundary conditions present.
3. What is a point heat source in the context of the steady state heat equation?
A point heat source is a localized area within a system that is generating or absorbing heat. In the steady state heat equation, a point heat source is represented by a single point within the system
where the temperature is known or can be calculated.
4. How is the steady state heat equation solved for a rectangle with a point heat source?
The steady state heat equation can be solved using various numerical methods, such as finite difference or finite element methods. These methods involve discretizing the rectangle into smaller
elements and solving the resulting equations to determine the temperature distribution in the system, taking into account the point heat source and any other relevant factors.
5. What are some real-world applications of the steady state heat equation in a rectangle with a point heat source?
The steady state heat equation has many practical applications, such as in the design of heating and cooling systems in buildings, the analysis of heat transfer in electronic devices, and the study
of thermal properties in materials. It can also be used in the field of geothermal energy to model the flow of heat in the Earth's crust.
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