text:
# Heat exchanger

Author: Alex Valdes [2015], Cindy Chen [2016]

Stewards: Jian Gong, and Fengqi You

## Introduction

Heat exchangers are necessary process units that are part of any detailed process flow diagram. Process streams commonly interact through heat exchangers in order to save money on heating and cooling utilities. Furthermore, the surface area of the heat exchanger is proportional to the amount of heat that can be transferred and is the most indicative cost component of a heat exchanger (Wilcox, 2009). Therefore, all of the commercial simulators include models for heaters, coolers, heat exchangers, fired heaters,and air coolers (Towler and Sinnott, 2013). Typically, the only inputs necessary for heat exchanger models to converge are properly specified inlet streams (flow rate, temperature, pressure, composition), the pressure drop of flow pathways, and the outlet temperatures or the duty.

# Aspen HYSYS V8.0

Model simulators such as HYSYS are extremely useful for engineers to quickly estimate capital costs and utility requirements.

### Model Equations

HYSYS uses the following equations for adiabatic steady state heat exchangers (Wilcox, 2009):

$Q = Nps(Hps,in-Hps,out)$ (heat transferred from process stream)

$Q = Nus(Hus,out-Hus,in)$ (heat transferred to utility stream)

$Q = UAF(dTavg)$ (rate of heat transfer)

where: Q is the rate of heat exchange (e.g., in kJ/h), Ni is the flowrate of stream i (e.g, in kmol/h), Hi is the specific enthalpy of stream i (kJ/kmol), U is the overall heat transfer coefficient (kJ/m2.K), A is the heat exchange area (m2), F is the correction factor for the deviation from cocurrent or countercurrent flow, dTavg is the average temperature difference between the streams for true cocurrent or countercurrent flow.

### Heuristics

Before performing simulation, it is important to have an idea of practical considerations and parameter values. A comprehensive list of Heuristics in Chemical Engineering (Walas, 1990) is found at http://people.clarkson.edu/~wwilcox/Design/heurist.pdf. For shell and tube heat exchangers:

• Tube side is for corrosive, fouling, scaling, and high pressure fluids.
• Shell side is for viscous and condensing fluids.
• Pressure drops are 1.5 psi for boiling and 3-9 psi for other services.
• Water inlet temperature is 90°F, maximum outlet 120°F.

More rules of thumb for various types of heat exchangers, as well as many other pieces of process equipment, are contained in the source above.

### Types of Heat Transfer Operations in HYSYS

Although shell and tube heat exchangers are widely used in practice, and most commonly taught to engineers first learning heat transfer, HYSYS contains various heat transfer units:

Table 1: HYSYS Heat Transfer Simulation Operations

Unit Heat Exchange Between Input(s) Output(s)
Cooler Hot process stream and utility (energy) Flow rate, composition of inlet stream. For inlet and outlet stream TWO out of THREE: temperature, pressure, vapor phase fraction Duty, Q
Heater Cold process stream and utility (energy) Same as cooler Duty, Q
Heat Exchanger Two process streams, shell and tube geometry Same as cooler for first process stream. For second process stream: composition, and TWO out of THREE: temperature, pressure, vapor phase fraction for inlet and outlet Second process stream flow rate; Duty, Q; Heat Transfer Coefficient, U
LNG Exchanger Multiple process streams, plate or plate-fin geometry Same as shell and tube Heat Exchanger Same as shell and tube Heat Exchanger
Air Cooler Process stream and ideal air mixture Same as cooler for process stream. Total air flow; Fan rating; Heat transfer coefficient, U
Fired Heater (Furnace) Combustion fuel gas and process stream(s) Same as cooler for process stream. Air/fuel ratio Flow rate of fuel/air; Duty, Q; Heat transfer coefficient, U

## Tutorial for Heat Exchanger

#### Input Properties

• Open Aspen HYSYS and create a New Case under File menu.
• Create a component list by adding all components present in the process.
• Select a thermodynamic fluid package that is applicable to the process see Property Package wiki article for more details on options or go to http://people.clarkson.edu/~wwilcox/Design/thermodl.htm

#### Enter Simulation

• In the model palette, there are a few options for heating/cooling units. Use a Heater or a Cooler to change the temperature of one process stream using a utility. Use a Heat Exchanger for two process streams exchanging heat, thereby changing the temperature of each. The following steps are for a Heat Exchanger, but the steps are similar for a Heater or Cooler.
• Click on the exchanger in the Flowsheet window. In the Design Tab under Connections, create Tube and Shell Side Inlet and Outlet streams (four streams). Choosing which fluid goes tube side and which goes shell depends on many factors, but some rules of thumb include putting high pressure and/or corrosive fluids tube side (Sloley, 2013).
• To get a feel for how HYSYS interprets the situation, open the Specs section still in the Design Tab (image to the right). It can be seen that HYSYS needs five pieces of information (Degrees of Freedom =5) about the inlet and outlet streams. If the process is overspecified, HYSYS should give a message indicating the type of error, or may simply say no solution. After understanding the information in this table, enter four inlet parameters and one outlet parameter (the temperature of one of the outlets).
• At this point, HYSYS should say that there is an unknown Delta P. Specify (guess) the inlet and outlet pressures, the exchanger should then converge. Good initial guesses for pressure drop are between 0.3-0.7 bar or 30-70 kPa (Towler and Sinnott, 2013)

Figure 1. Heat Exchanger in Flowsheet and Design Specs Page

#### Customizing Heat Exchanger

• Once the exchanger has converged and the desired outlet temperature of one of the process streams has been achieved, various parameters can modified and more detailed analyses on the exchanger can be looked at.
• Under the Design Tab, under Parameters, choose one of the five potential heat exchanger models (most common choice is rigorous shell and tube).

Troubleshooting note: sometimes the rigorous heat exchanger model won't converge at first, even if the system is properly specified. Sometimes when this happens, switching to Simple End Point model and then switching back to the rigorous model will make it converge.

• At this point, utilize a neat feature of HYSYS in the Rigorous Model Section of the Parameters page: Size Rigorous Shell&Tube. Clicking on this will have HYSYS automatically run cases to determine optimal parameters of pressure drop and heat exchanger area (there is also an option to automatically run this feature). Once this feature converges, very useful information about the required overall heat transfer coefficient, pressure drop, heat transfer area, and other physical parameters is readily viewable under the Rating Tab.
• Also under the Rating tab, physical features of the heat exchanger such as the number of shell/tube passes can be modified (real heat exchangers typically have more than one or two tube passes per shell). These features of course change the necessary heat transfer coefficient (favorably), pressure drop (unfavorably), and unspecified outlet stream conditions.

Figure 2: Rating Tab containing physical/geometric parameters of exchanger

### Common Issues

There a few common problems that arise when using heat exchangers in these programs.

When heat exchangers are used with streams that go to earlier stages of the process, an information loop occurs and the program is less likely to converge. Many times the process design requires a later process stream, such as the bottoms of a distillation column, to heat an earlier stream, such as the feed to the same column (Towler and Sinnott, 2012).

Another problem arises if the specifications of the heat exchanger are impossible, but the model still converges with physically unreasonable results - such as a temperature cross.

To avoid these problems, it is good practice to use utility heaters and coolers instead of heat exchangers to get an idea of the required heat load and parameters of an exchanger. The heaters and coolers are also useful for obtaining initial guesses of outlet temperatures and pressure drops. After that information is obtained, the designer will have a much better chance of simulating a heat exchanger that will converge with meaningful results (Wilxcox, 2009).

# Heat Exchanger Networks

## Introduction

A typical plant requires the use of many individual heat exchanger units. This combined process heating and cooling of a plant could be accomplished through the use of utilities, but this will be very costly in term of capital and utility costs. Heat exchanger networks utilize multiple heat exchanger units to optimize the process design. Effective incorporation of heat exchanger networks will minimize the use of utilities, minimize the number of heat exchangers, and minimize costs associated with the design.

## Model Equations and Heuristics

Heat exchanger networks are designed through insightful use of pinch analysis. The pinch point divides the temperature range into two regions. Detailed calculations on obtaining the pinch point can be found on the pinch analysis page.

A valid heat exchanger network system must follow appropriate heuristics. Heating utility can only be used above the pinch, while cooling utility can only be used below it. Additionally, heat from a hot stream with temperature above the pinch must not be transferred to a cold stream with temperature below the pinch (Towler and Sinnot, 2012).

## Minimum Energy Recovery Targets

The primary objective of building heat exchanger networks is to maximize the energy recovery from the hot process streams to the cold process streams. Thus, before starting the synthesis of a heat exchanger network, minimum energy recovery (MER) targets must be determined. There are three methods to doing so: (1) the temperature-interval method, (2) a graphical method using composite heating and cooling curves, and (3) linear programming (Seider, 2004). The temperature-interval and composite curve methods have been outlined in pinch analysis page. A linear programming example is discussed below.

### Linear Programming

Consider a system where two cold streams, C1 and C2, are to be heated and two hot streams, H1 and H2, are to be cooled without phase change. Ths conditions and properties of these streams are listed below.

Stream Initial Temp. °F Target Temp. °F Heat Capacity Flow Rate, Btu/hr°F Heat Load, Btu/hr
C1 120 235 20,000 230
C2 180 240 40,000 240
H1 260 160 30,000 300
H2 250 130 15,000 180

The linear program can be formulated as:

```   Minimize Q_steam
with respect to (w.r.t.):
Q_steam
Subject to (s.t.):

```

This example is taken from Process Design Pinciples: Synthesis, Analysis, and Evaluation by Seider, Seader, and Lewin.

## Future Developments

While heat exchanger networks try to optimize the energy costs for heating and cooling utility, capital costs are often difficult to calculate and incorporate in the analysis. Around the pinch point, the hot and cold streams are most constrained. This leads to the requirement of large heat exchangers to effectively transfer heat between hot and cold streams. A design for maximum heat recovery and minimum number of exchangers will typically include a loop in the network.

Example of loop in network design (Towler and Sinnot, 2012)

As seen in the figure above, a loop may cross the pinch point. These loops can be broken by transferring heat across the pinch point. This, however, violates the heuristic that heat should not be transferred across the pinch point and at least one violation of the specific ΔT_min will also occur. To restore ΔT_min, heat can be shifted along a path, which increases the energy consumption of the process. From this dilemma, it is clear that there are pros and cons to the attempt to minimize any heat exchanger network.

Analysis with the incorporation of capital costs can be done by assessing the heat exchanger network as a mixed integer non-linear programming (MINLP) problem. MINLP problems are typically solved using numerical solvers. But large scale MINLP problems, such as the ones that may be generated for a process plant, may be difficult to solve using available numerical algorithms and solvers.

# References

1. G.P. Towler, R. Sinnott. Chemical Engineering Design: Principles, Practice and Economics of Plant and Process Design. Elsevier, 2012.
2. Sloley. "Shell-and-Tube Heat Exchanger: Pick the Right Side". Chemical Processing. October, 2013
3. Wilcox. "Sizing of heat exchangers using HYSYS or UniSim". 2009. http://people.clarkson.edu/~wwilcox/Design/hxsizing.htm date accessed: March 1, 2015
4. Walas, S.M.. (1990). Chemical Process Equipment - Selection and Design. Elsevier


text:
# Meteorology - radiative equilibrium timescale

by Sojourner01
Tags: equilibrium, meteorology, radiative, timescale
 P: 373 1. The problem statement, all variables and given/known data Derive the radiative timescale for an atmosphere: $$\tau_{E} = \frac{c_{p} p_{0}}{4 g \sigma T^{3}_{E}}$$ 2. Relevant equations As above 3. The attempt at a solution I've gathered that the difference between the radiative power of an object, $$\sigma (T + \Delta T)^{4}$$ And the incoming solar flux on the object, $$(1 - \sigma) S$$, is equal to an instantaneous rate of change of heat, $$\frac{dQ}{dt}$$. I don't know how to proceed from here; my derivation of the answer doesn't appear to conform to the one above. edit: oh for crying out loud, I hate TeX. It never does what I want it to, and I have the 'how to program tex' thread open here in front of me. You can see what I was trying to achieve.
 P: 373 Well, I've sorted it out in case anyone is interested. Turns out you can factor out the bracket $$(T + \Delta T)^{4}$$ as $$T^{4}(1 + \frac{4 \Delta T}{T})$$ using the first order binomial expansion. Silly method, but there we go.

 Related Discussions Forum Feedback & Announcements 2 General Astronomy 3 General Physics 2 Forum Feedback & Announcements 2 Earth 2

format: plain


text:
# zbMATH — the first resource for mathematics

On real valued $$I$$-convergent a-summable sequence spaces defined by sequences of Orlicz functions. (English) Zbl 1340.46004
Summary: We introduce some new sequence spaces using $$I$$-convergence and sequences of Orlicz functions and study some basic topological and algebraic properties of these spaces. Also, we investigate the relations between these spaces.

##### MSC:
 46A45 Sequence spaces (including Köthe sequence spaces) 40A05 Convergence and divergence of series and sequences 40C05 Matrix methods for summability
##### Keywords:
$$I$$-convergence; Orlicz function; matrix transformation


text:
## First observation and measurement of the branching fraction for the decay B-s(0) -> D-s*K--/+(+/-)

The first observation of the B-s(0) -> D-s*(-/+) K-+/- decay is reported using 3.0 fb(-1) of proton-proton collision data collected by the LHCb experiment. The D-s*(-/+) mesons are reconstructed through the decay chain D-s*(-/+) -> gamma D-s(-/+) ((KK +/-)-K--/+pi(-/+)). The branching fraction relative to that for B-s(0) -> D-s*(-)pi(+) decays is measured to be B (B-s(0) -> D-s*K--/+(+/-))/B(B-s(0) -> D-s*(-)pi(+)) = 0.068 +/- 0.005(-0.002)(+0.003), where the first uncertainty is statistical and the second is systematic. Using a recent measurement of B(B-s(0) -> D-s*(-)pi(+)), the absolute branching fraction of B-s(0) -> Ds*K--/+(+/-) is measured as B(B-s(0) -> D*K--/+(+/-)) = (16.3 +/- 1.2(stat)(-0.5)(+0.7)(syst) +/- 4.8(norm)) x 10(-5), where the third uncertainty is due to the uncertainty on the branching fraction of the normalisation channel.

Published in:
Journal of High Energy Physics, 06, 130
Year:
2015
Publisher:
New York, Springer
ISSN:
1029-8479
Keywords:
Laboratories:


text:
Tersian, Stepan Agop

Compute Distance To:
 Author ID: tersian.stepan-agop Published as: Tersian, Stepan; Tersian, Stepan A.; Tersian, S.; Tersian, S. A.; Tersian, Stepan Agop; Tersian, St. A. more...less
 Documents Indexed: 79 Publications since 1982, including 3 Books 2 Contributions as Editor Reviewing Activity: 425 Reviews Co-Authors: 46 Co-Authors with 63 Joint Publications 1,223 Co-Co-Authors
all top 5

Co-Authors

 17 single-authored 8 Cabada, Alberto 5 Chaparova, Julia V. 5 Li, Lin 4 do Rosário Grossinho, Maria 4 Souroujon, Diko M. 3 Aprahamian, Meline Onik 3 Dimitrov, Nikolay D. 3 Gyulov, Tihomir 3 Iannizzotto, Antonio 3 Mihăilescu, Mihai 3 Moroşanu, Gheorghe 3 Rădulescu, Vicenţiu D. 3 Saavedra, Lorena 2 Biroli, Marco 2 Bonanno, Gabriele 2 Drábek, Pavel 2 Heidarkhani, Shapour 2 Kelevedjiev, Petio S. 2 Khristova, Snezhana G. 2 Kristály, Alexandru 2 Peletier, Lambertus Adrianus 2 Rodríguez-López, Rosana 2 Stavroulakis, Ioannis P. 2 Zabreĭko, Pëtr Petrovich 1 Almeida, Ricardo 1 Averna, Diego 1 Caristi, Giuseppe 1 Chinnì, Antonia 1 D’Aguì, Giuseppina 1 Di Bella, Beatrice 1 Grammatikopoulos, Myron K. 1 Hadjian, Armin 1 Khaleghi Moghadam, M. 1 Langerová, Martina 1 Li, Chengyue 1 Minhós, Feliz Manuel 1 Moradi, Shahin 1 Nastasi, Antonella 1 Nyamoradi, Nemat 1 Popivanov, Petar Radoev 1 Precup, Radu 1 Salari, Amjad 1 Sanchez, Luís 1 Sun, Jijiang 1 Tornatore, Elisabetta 1 Vetro, Calogero
all top 5

Serials

 7 Journal of Mathematical Analysis and Applications 6 Doklady Bolgarskoĭ Akademii Nauk 6 Fractional Calculus & Applied Analysis 5 Electronic Journal of Differential Equations (EJDE) 4 Rendiconti. Scienze Matemàtiche e Applicazioni. A 4 Applied Mathematics Letters 4 Electronic Journal of Qualitative Theory of Differential Equations 3 Nonlinear Analysis. Theory, Methods & Applications 2 Applied Mathematics and Computation 2 Topological Methods in Nonlinear Analysis 2 Nonlinear Analysis. Real World Applications 2 Boundary Value Problems 1 Applicable Analysis 1 Mathematical Methods in the Applied Sciences 1 Demonstratio Mathematica 1 Funkcialaj Ekvacioj. Serio Internacia 1 Journal of Differential Equations 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Results in Mathematics 1 Bulletin of the Iranian Mathematical Society 1 Dynamic Systems and Applications 1 Turkish Journal of Mathematics 1 Advances in Differential Equations 1 Journal of Difference Equations and Applications 1 Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 1 Communications in Contemporary Mathematics 1 Discrete and Continuous Dynamical Systems. Series B 1 Portugaliae Mathematica. Nova Série 1 Acta Mathematica Scientia. Series B. (English Edition) 1 Advances in Difference Equations 1 Nonconvex Optimization and Its Applications 1 Mathematics and Natural Sciences 1 Open Mathematics
all top 5

Fields

 45 Ordinary differential equations (34-XX) 30 Partial differential equations (35-XX) 22 Global analysis, analysis on manifolds (58-XX) 14 Operator theory (47-XX) 13 Difference and functional equations (39-XX) 5 Calculus of variations and optimal control; optimization (49-XX) 4 Dynamical systems and ergodic theory (37-XX) 4 Functional analysis (46-XX) 3 Integral equations (45-XX) 3 Numerical analysis (65-XX) 2 General and overarching topics; collections (00-XX) 2 Measure and integration (28-XX) 2 Systems theory; control (93-XX) 1 Real functions (26-XX) 1 Potential theory (31-XX) 1 Mechanics of particles and systems (70-XX) 1 Mechanics of deformable solids (74-XX) 1 Biology and other natural sciences (92-XX) 1 Mathematics education (97-XX)

Citations contained in zbMATH Open

59 Publications have been cited 619 times in 443 Documents Cited by Year
Existence of solutions to boundary value problem for impulsive fractional differential equations. Zbl 1308.34010
Bonanno, Gabriele; Rodríguez-López, Rosana; Tersian, Stepan
2014
Multiple solutions for discrete boundary value problems. Zbl 1169.39008
Cabada, Alberto; Iannizzotto, Antonio; Tersian, Stepan
2009
Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations. Zbl 0984.34031
Tersian, Stepan; Chaparova, Julia
2001
Eigenvalue problems for anisotropic discrete boundary value problems. Zbl 1181.47016
Mihăilescu, Mihai; Rădulescu, Vicenţiu; Tersian, Stepan
2009
Multiple solutions to boundary value problem for impulsive fractional differential equations. Zbl 1312.34024
Rodríguez-López, Rosana; Tersian, Stepan
2014
Multiple homoclinic solutions for the discrete $$p$$-Laplacian via critical point theory. Zbl 1282.39003
Iannizzotto, Antonio; Tersian, Stepan A.
2013
Multiplicity of solutions of a two point boundary value problem for a fourth-order equation. Zbl 1294.34016
2013
The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation. Zbl 0960.34013
Grossinho, M. R.; Tersian, St. A.
2000
On the solvability of a boundary value problem for a fourth-order ordinary differential equation. Zbl 1087.34508
Grossinho, Maria do Rosário; Sanchez, Luis; Tersian, Stepan A.
2005
Homoclinic solutions of difference equations with variable exponents. Zbl 1270.39005
Mihăilescu, Mihai; Rădulescu, Vicenţiu D.; Tersian, Stepan
2011
On the existence and multiplicity of solutions for Dirichlet’s problem for fractional differential equations. Zbl 1337.34009
Averna, Diego; Tersian, Stepan; Tornatore, Elisabetta
2016
An introduction to minimax theorems and their applications to differential equations. Zbl 0987.58004
Grossinho, Maria do Rosário; Tersian, Stepan Agop
2001
A minimax theorem and applications to nonresonance problems for semilinear equations. Zbl 0605.49007
Tersian, Stepan A.
1986
Multiplicity results for superlinear boundary value problems with impulsive effects. Zbl 1342.34046
D’Aguì, Giuseppina; Di Bella, Beatrice; Tersian, Stepan
2016
Existence and nonexistence of nontrivial solutions of semilinear fourth- and sixth-order differential equations. Zbl 1126.34316
Chaparova, Julia V.; Peletier, Lambertus A.; Tersian, Stepan A.
2003
On homoclinic solutions of a semilinear $$p$$-Laplacian difference equation with periodic coefficients. Zbl 1207.39004
Cabada, Alberto; Li, Chengyue; Tersian, Stepan
2010
Quasilinear elliptic problems in $$\mathbb R^N$$ involving oscillatory nonlinearities. Zbl 1185.35071
Kristály, Alexandru; Moroşanu, Gheorghe; Tersian, Stepan
2007
Decreasing and fast solutions for a second-order difference equation related to Fisher-Kolmogorov’s equation. Zbl 1257.39008
Aprahamian, Meline; Souroujon, Diko; Tersian, Stepan
2010
Existence and nonexistence of nontrivial solutions of semilinear sixth-order ordinary differential equations. Zbl 1122.34305
Chaparova, J. V.; Peletier, L. A.; Tersian, S. A.
2004
Spectral estimates for a nonhomogeneous difference problem. Zbl 1215.47020
Kristály, Alexandru; Mihăilescu, Mihai; Rădulescu, Vicenţiu; Tersian, Stepan
2010
Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations. Zbl 1319.34044
2014
Existence results for a two point boundary value problem involving a fourth-order equation. Zbl 1349.34070
Bonanno, Gabriele; Chinnì, Antonia; Tersian, Stepan
2015
Existence of heteroclinic solutions for discrete $$p$$-Laplacian problems with a parameter. Zbl 1235.39002
2011
On symmetric positive homoclinic solutions of semilinear $$p$$-Laplacian differential equations. Zbl 1281.34032
Tersian, Stepan
2012
Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian. Zbl 1377.35099
Li, Lin; Sun, Jijiang; Tersian, Stepan
2017
Multiple positive solutions to a fourth-order boundary-value problem. Zbl 1357.34054
2016
Periodic and homoclinic solutions of some semilinear sixth-order differential equations. Zbl 1017.34046
Tersian, Stepan; Chaparova, Julia
2002
Existence of trivial and nontrivial solutions of fourth-order ordinary differential equation. Zbl 1065.34016
Gyulov, T.; Tersian, S.
2004
Singular and nonsingular first-order initial value problems. Zbl 1191.34010
Kelevedjiev, P. S.; Tersian, S.
2010
Existence for a semilinear sixth-order ODE. Zbl 1106.34007
Gyulov, Tihomir; Morosanu, Gheorghe; Tersian, Stepan
2006
Existence of solutions for nonlinear fractional order $$p$$-Laplacian differential equations via critical point theory. Zbl 1481.34089
2019
Multiple solutions for degenerate nonlocal problems. Zbl 06892636
Caristi, Giuseppe; Heidarkhani, Shapour; Salari, Amjad; Tersian, Stepan A.
2018
Existence of solutions for $$2n$$th-order nonlinear $$p$$-Laplacian differential equations. Zbl 1354.34045
Saavedra, Lorena; Tersian, Stepan
2017
Three solutions for second-order boundary-value problems with variable exponents. Zbl 1413.34098
Heidarkhani, Shapour; Moradi, Shahin; Tersian, Stepan
2018
Partial differential equations. An introduction with Mathematica and MAPLE. Zbl 0940.35001
Stavroulakis, Ioannis P.; Tersian, Stepan A.
1999
Scalar linear impulsive Riemann-Liouville fractional differential equations with constant delay-explicit solutions and finite time stability. Zbl 1450.34056
Hristova, Snezhana G.; Tersian, Stepan A.
2020
Fractional problems with critical nonlinearities by a sublinear perturbation. Zbl 1448.35195
Li, Lin; Tersian, Stepan
2020
Existence of three solutions for a discrete anisotropic boundary value problem. Zbl 1408.39002
Khaleghi Moghadam, M.; Li, L.; Tersian, S.
2018
On the solvability of semilinear Schrödinger equations in strip-like domains. Zbl 0937.35031
Tersian, S.
1998
Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations. Zbl 0969.34038
Tersian, Stepan; Chaparova, Julia
2000
Existence of trivial and nontrivial solutions of a fourth-order ordinary differential equation. Zbl 1058.34016
Gyulov, Tihomir; Tersian, Stepan
2004
Heteroclinic solutions of a second-order difference equation related to the Fisher-Kolmogorov’s equation. Zbl 1245.39003
Cabada, Alberto; Souroujon, Diko; Tersian, Stepan
2012
Characterizations of the range of Neumann problem for semilinear elliptic equations. Zbl 0652.35040
Drábek, Pavel; Tersian, Stepan A.
1987
On the periodic problem for the equation $$x''(t)+g(x(t))=f(t)$$. Zbl 0579.34031
Tersian, Stepan
1985
Partial differential equations. An introduction with Mathematica and MAPLE. 2nd ed. Zbl 1062.35001
Stavroulakis, Ioannis P.; Tersian, Stepan A.
2004
Existence of solutions for nonlinear $$p$$-Laplacian difference equations. Zbl 06850994
Saavedra, Lorena; Tersian, Stepan
2017
Existence results for a mixed boundary value problem. Zbl 1349.34056
2015
Existence of homoclinic solutions for a nonlinear fourth order $$p$$-Laplacian difference equation. Zbl 1444.39013
Dimitrov, Nikolay; Tersian, Stepan
2020
Homoclinic and periodic solutions for some classes of second order differential equations. Zbl 0997.34034
Grossinho, M. R.; Minhós, F.; Tersian, S.
2001
Nontrivial solutions of semilinear Schrödinger equations on $$\mathbb{R}^ n$$ and strip-like domains. Zbl 0832.35045
Tersian, Stepan
1995
A note on Palais-Smale condition and mountain-pass principle for locally Lipschitz functionals. Zbl 0857.49005
Tersian, Stepan
1995
Mountain pass theorems, deformation theorems, and Palais-Smale conditions. Zbl 1053.58006
Tersian, Stepan A.
2001
On a mini max theorem. Zbl 0565.49008
Tersian, S. A.
1985
Existence and multiplicity of periodic solutions to one-dimensional $$p$$-Laplacian. Zbl 1363.34109
Drabek, Pavel; Langerova, Martina; Tersian, Stepan
2016
Homoclinic solutions of a fourth-order travelling wave ODE. Zbl 1162.34036
Morosanu, Gheorghe; Souroujon, Diko; Tersian, Stepan
2007
On the existence of nontrivial solutions to a semilinear equation relative to a Dirichlet form. Zbl 1138.35324
Biroli, Marco; Tersian, Stepan
1998
Hammerstein integral equations with nontrivial solutions. Zbl 0721.45002
Tersian, Stepan A.; Zabrejko, Petr P.
1991
On a class of abstract systems without resonance in a Hilbert space. Zbl 0489.35007
Tersian, Stepan A.
1982
Existence of solutions of discrete equations via critical point theory. Zbl 1242.39003
Cabada, Alberto; Iannizzotto, Antonio; Tersian, Stepan
2011
Scalar linear impulsive Riemann-Liouville fractional differential equations with constant delay-explicit solutions and finite time stability. Zbl 1450.34056
Hristova, Snezhana G.; Tersian, Stepan A.
2020
Fractional problems with critical nonlinearities by a sublinear perturbation. Zbl 1448.35195
Li, Lin; Tersian, Stepan
2020
Existence of homoclinic solutions for a nonlinear fourth order $$p$$-Laplacian difference equation. Zbl 1444.39013
Dimitrov, Nikolay; Tersian, Stepan
2020
Existence of solutions for nonlinear fractional order $$p$$-Laplacian differential equations via critical point theory. Zbl 1481.34089
2019
Multiple solutions for degenerate nonlocal problems. Zbl 06892636
Caristi, Giuseppe; Heidarkhani, Shapour; Salari, Amjad; Tersian, Stepan A.
2018
Three solutions for second-order boundary-value problems with variable exponents. Zbl 1413.34098
Heidarkhani, Shapour; Moradi, Shahin; Tersian, Stepan
2018
Existence of three solutions for a discrete anisotropic boundary value problem. Zbl 1408.39002
Khaleghi Moghadam, M.; Li, L.; Tersian, S.
2018
Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian. Zbl 1377.35099
Li, Lin; Sun, Jijiang; Tersian, Stepan
2017
Existence of solutions for $$2n$$th-order nonlinear $$p$$-Laplacian differential equations. Zbl 1354.34045
Saavedra, Lorena; Tersian, Stepan
2017
Existence of solutions for nonlinear $$p$$-Laplacian difference equations. Zbl 06850994
Saavedra, Lorena; Tersian, Stepan
2017
On the existence and multiplicity of solutions for Dirichlet’s problem for fractional differential equations. Zbl 1337.34009
Averna, Diego; Tersian, Stepan; Tornatore, Elisabetta
2016
Multiplicity results for superlinear boundary value problems with impulsive effects. Zbl 1342.34046
D’Aguì, Giuseppina; Di Bella, Beatrice; Tersian, Stepan
2016
Multiple positive solutions to a fourth-order boundary-value problem. Zbl 1357.34054
2016
Existence and multiplicity of periodic solutions to one-dimensional $$p$$-Laplacian. Zbl 1363.34109
Drabek, Pavel; Langerova, Martina; Tersian, Stepan
2016
Existence results for a two point boundary value problem involving a fourth-order equation. Zbl 1349.34070
Bonanno, Gabriele; Chinnì, Antonia; Tersian, Stepan
2015
Existence results for a mixed boundary value problem. Zbl 1349.34056
2015
Existence of solutions to boundary value problem for impulsive fractional differential equations. Zbl 1308.34010
Bonanno, Gabriele; Rodríguez-López, Rosana; Tersian, Stepan
2014
Multiple solutions to boundary value problem for impulsive fractional differential equations. Zbl 1312.34024
Rodríguez-López, Rosana; Tersian, Stepan
2014
Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations. Zbl 1319.34044
2014
Multiple homoclinic solutions for the discrete $$p$$-Laplacian via critical point theory. Zbl 1282.39003
Iannizzotto, Antonio; Tersian, Stepan A.
2013
Multiplicity of solutions of a two point boundary value problem for a fourth-order equation. Zbl 1294.34016
2013
On symmetric positive homoclinic solutions of semilinear $$p$$-Laplacian differential equations. Zbl 1281.34032
Tersian, Stepan
2012
Heteroclinic solutions of a second-order difference equation related to the Fisher-Kolmogorov’s equation. Zbl 1245.39003
Cabada, Alberto; Souroujon, Diko; Tersian, Stepan
2012
Homoclinic solutions of difference equations with variable exponents. Zbl 1270.39005
Mihăilescu, Mihai; Rădulescu, Vicenţiu D.; Tersian, Stepan
2011
Existence of heteroclinic solutions for discrete $$p$$-Laplacian problems with a parameter. Zbl 1235.39002
2011
Existence of solutions of discrete equations via critical point theory. Zbl 1242.39003
Cabada, Alberto; Iannizzotto, Antonio; Tersian, Stepan
2011
On homoclinic solutions of a semilinear $$p$$-Laplacian difference equation with periodic coefficients. Zbl 1207.39004
Cabada, Alberto; Li, Chengyue; Tersian, Stepan
2010
Decreasing and fast solutions for a second-order difference equation related to Fisher-Kolmogorov’s equation. Zbl 1257.39008
Aprahamian, Meline; Souroujon, Diko; Tersian, Stepan
2010
Spectral estimates for a nonhomogeneous difference problem. Zbl 1215.47020
Kristály, Alexandru; Mihăilescu, Mihai; Rădulescu, Vicenţiu; Tersian, Stepan
2010
Singular and nonsingular first-order initial value problems. Zbl 1191.34010
Kelevedjiev, P. S.; Tersian, S.
2010
Multiple solutions for discrete boundary value problems. Zbl 1169.39008
Cabada, Alberto; Iannizzotto, Antonio; Tersian, Stepan
2009
Eigenvalue problems for anisotropic discrete boundary value problems. Zbl 1181.47016
Mihăilescu, Mihai; Rădulescu, Vicenţiu; Tersian, Stepan
2009
Quasilinear elliptic problems in $$\mathbb R^N$$ involving oscillatory nonlinearities. Zbl 1185.35071
Kristály, Alexandru; Moroşanu, Gheorghe; Tersian, Stepan
2007
Homoclinic solutions of a fourth-order travelling wave ODE. Zbl 1162.34036
Morosanu, Gheorghe; Souroujon, Diko; Tersian, Stepan
2007
Existence for a semilinear sixth-order ODE. Zbl 1106.34007
Gyulov, Tihomir; Morosanu, Gheorghe; Tersian, Stepan
2006
On the solvability of a boundary value problem for a fourth-order ordinary differential equation. Zbl 1087.34508
Grossinho, Maria do Rosário; Sanchez, Luis; Tersian, Stepan A.
2005
Existence and nonexistence of nontrivial solutions of semilinear sixth-order ordinary differential equations. Zbl 1122.34305
Chaparova, J. V.; Peletier, L. A.; Tersian, S. A.
2004
Existence of trivial and nontrivial solutions of fourth-order ordinary differential equation. Zbl 1065.34016
Gyulov, T.; Tersian, S.
2004
Existence of trivial and nontrivial solutions of a fourth-order ordinary differential equation. Zbl 1058.34016
Gyulov, Tihomir; Tersian, Stepan
2004
Partial differential equations. An introduction with Mathematica and MAPLE. 2nd ed. Zbl 1062.35001
Stavroulakis, Ioannis P.; Tersian, Stepan A.
2004
Existence and nonexistence of nontrivial solutions of semilinear fourth- and sixth-order differential equations. Zbl 1126.34316
Chaparova, Julia V.; Peletier, Lambertus A.; Tersian, Stepan A.
2003
Periodic and homoclinic solutions of some semilinear sixth-order differential equations. Zbl 1017.34046
Tersian, Stepan; Chaparova, Julia
2002
Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations. Zbl 0984.34031
Tersian, Stepan; Chaparova, Julia
2001
An introduction to minimax theorems and their applications to differential equations. Zbl 0987.58004
Grossinho, Maria do Rosário; Tersian, Stepan Agop
2001
Homoclinic and periodic solutions for some classes of second order differential equations. Zbl 0997.34034
Grossinho, M. R.; Minhós, F.; Tersian, S.
2001
Mountain pass theorems, deformation theorems, and Palais-Smale conditions. Zbl 1053.58006
Tersian, Stepan A.
2001
The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation. Zbl 0960.34013
Grossinho, M. R.; Tersian, St. A.
2000
Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations. Zbl 0969.34038
Tersian, Stepan; Chaparova, Julia
2000
Partial differential equations. An introduction with Mathematica and MAPLE. Zbl 0940.35001
Stavroulakis, Ioannis P.; Tersian, Stepan A.
1999
On the solvability of semilinear Schrödinger equations in strip-like domains. Zbl 0937.35031
Tersian, S.
1998
On the existence of nontrivial solutions to a semilinear equation relative to a Dirichlet form. Zbl 1138.35324
Biroli, Marco; Tersian, Stepan
1998
Nontrivial solutions of semilinear Schrödinger equations on $$\mathbb{R}^ n$$ and strip-like domains. Zbl 0832.35045
Tersian, Stepan
1995
A note on Palais-Smale condition and mountain-pass principle for locally Lipschitz functionals. Zbl 0857.49005
Tersian, Stepan
1995
Hammerstein integral equations with nontrivial solutions. Zbl 0721.45002
Tersian, Stepan A.; Zabrejko, Petr P.
1991
Characterizations of the range of Neumann problem for semilinear elliptic equations. Zbl 0652.35040
Drábek, Pavel; Tersian, Stepan A.
1987
A minimax theorem and applications to nonresonance problems for semilinear equations. Zbl 0605.49007
Tersian, Stepan A.
1986
On the periodic problem for the equation $$x''(t)+g(x(t))=f(t)$$. Zbl 0579.34031
Tersian, Stepan
1985
On a mini max theorem. Zbl 0565.49008
Tersian, S. A.
1985
On a class of abstract systems without resonance in a Hilbert space. Zbl 0489.35007
Tersian, Stepan A.
1982
all top 5

Cited by 505 Authors

 22 Tersian, Stepan Agop 20 Heidarkhani, Shapour 14 Caristi, Giuseppe 12 Moradi, Shahin 12 O’Regan, Donal 10 Cabada, Alberto 10 Galewski, Marek 9 Afrouzi, Ghasem Alizadeh 9 Bonanno, Gabriele 9 Kong, Lingju 9 Ouaro, Stanislas 8 Guiro, Aboudramane 8 Minhós, Feliz Manuel 8 Zhao, Yulin 7 Figueiredo, Giovany Malcher 7 Kone, Blaise 7 Li, Lin 7 Ma, To Fu 7 Nyamoradi, Nemat 7 Salari, Amjad 6 Agarwal, Ravi P. 6 Chen, Haibo 6 Henderson, Johnny Lee 6 Kristály, Alexandru 6 Molica Bisci, Giovanni 6 Nieto Roig, Juan Jose 6 Pei, Minghe 6 Serban, Calin-Constantin 6 Sun, Juntao 6 Wang, Hui 6 Wieteska, Renata 5 Bai, Zhanbing 5 D’Aguì, Giuseppina 5 Di Bella, Beatrice 5 Graef, John R. 5 Ibrango, Idrissa 5 Jebelean, Petru 5 Li, Peiluan 5 Luca, Rodica 5 Moghadam, Mohsen Khaleghi 5 Moroşanu, Gheorghe 5 Rădulescu, Vicenţiu D. 5 Saoudi, Kamel 5 Zhou, Zhan 4 Arjunan, Mani Mallika 4 Gyulov, Tihomir 4 Khristova, Snezhana G. 4 Kong, Fanchao 4 Kuang, Juhong 4 Li, Chengyue 4 Li, Weiguo 4 Liu, Lishan 4 Ma, Ruyun 4 Repovš, Dušan D. 4 Rodríguez-López, Rosana 4 Wang, Libo 4 Zhang, Xingyong 3 Amster, Pablo 3 An, Yukun 3 Averna, Diego 3 Bereanu, Cristian 3 Chaparova, Julia V. 3 Chen, Pengyu 3 Cui, Yujun 3 Dai, Guowei 3 Daoues, Adel 3 dos Santos, Gelson C. G. 3 Ge, Bin 3 Hammami, Amani 3 Huang, Wenhua 3 Imbesi, Maurizio 3 Li, Yongkun 3 Li, Yongxiang 3 Li, Zheqing 3 Liu, Wenbin 3 Long, Yuhua 3 Mawhin, Jean L. 3 Mihăilescu, Mihai 3 Nascimento, Rúbia G. 3 Saavedra, Lorena 3 Sanchez, Luís 3 Shi, Haiping 3 Stegliński, Robert 3 Suganya, Selvaraj 3 Tornatore, Elisabetta 3 Torres Ledesma, César Enrique 3 Ur Rehman, Mujeeb 3 Wu, Tsungfang 3 Xiang, Mingqi 3 Yang, Liu 3 Yuan, Rong 3 Zhai, Chengbo 3 Zhang, Binlin 3 Zhang, Xuemei 3 Zhang, Ziheng 2 Alves, Claudianor Oliveira 2 Avci, Mustafa 2 Bai, Dingyong 2 Băleanu, Dumitru I. 2 Barilla, David ...and 405 more Authors
all top 5

Cited in 120 Serials

 34 Journal of Mathematical Analysis and Applications 33 Boundary Value Problems 24 Applied Mathematics Letters 24 Advances in Difference Equations 21 Applied Mathematics and Computation 21 Journal of Difference Equations and Applications 20 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 20 Fractional Calculus & Applied Analysis 13 Mediterranean Journal of Mathematics 11 Nonlinear Analysis. Real World Applications 9 Abstract and Applied Analysis 9 Journal of Function Spaces 6 Discrete Dynamics in Nature and Society 6 Qualitative Theory of Dynamical Systems 6 Journal of Applied Mathematics and Computing 6 Journal of Nonlinear Science and Applications 5 Applicable Analysis 5 Discrete and Continuous Dynamical Systems. Series B 5 Complex Variables and Elliptic Equations 5 Journal of Fixed Point Theory and Applications 5 AIMS Mathematics 4 Rocky Mountain Journal of Mathematics 4 Chaos, Solitons and Fractals 4 Journal of Differential Equations 4 Bulletin of the Iranian Mathematical Society 4 Nonlinear Analysis. Modelling and Control 4 Journal of Applied Mathematics 3 Computers & Mathematics with Applications 3 ZAMP. Zeitschrift für angewandte Mathematik und Physik 3 Journal of Computational and Applied Mathematics 3 Quaestiones Mathematicae 3 Opuscula Mathematica 3 Mathematical Problems in Engineering 3 Journal of Inequalities and Applications 3 International Journal of Nonlinear Sciences and Numerical Simulation 3 Cubo 3 Journal of Applied Analysis and Computation 2 Mathematical Methods in the Applied Sciences 2 Ukrainian Mathematical Journal 2 Proceedings of the American Mathematical Society 2 Acta Applicandae Mathematicae 2 Mathematical and Computer Modelling 2 Topological Methods in Nonlinear Analysis 2 NoDEA. Nonlinear Differential Equations and Applications 2 Discrete and Continuous Dynamical Systems 2 Arab Journal of Mathematical Sciences 2 Taiwanese Journal of Mathematics 2 Acta Mathematica Scientia. Series B. (English Edition) 2 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 2 Differential Equations and Applications 2 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 2 Journal of Mathematics 2 Nonautonomous Dynamical Systems 2 Open Mathematics 1 Journal of Fluid Mechanics 1 Mathematical Notes 1 Collectanea Mathematica 1 Czechoslovak Mathematical Journal 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of the Korean Mathematical Society 1 Journal of Optimization Theory and Applications 1 Manuscripta Mathematica 1 Mathematics and Computers in Simulation 1 Mathematische Nachrichten 1 Monatshefte für Mathematik 1 Siberian Mathematical Journal 1 Acta Mathematica Hungarica 1 Physica D 1 Applied Numerical Mathematics 1 Numerical Methods for Partial Differential Equations 1 Journal of Scientific Computing 1 Journal of Integral Equations and Applications 1 Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences 1 Applied Mathematical Modelling 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Expositiones Mathematicae 1 Indagationes Mathematicae. New Series 1 Continuum Mechanics and Thermodynamics 1 Tatra Mountains Mathematical Publications 1 Set-Valued Analysis 1 Applied Mathematics. Series B (English Edition) 1 Journal of Mathematical Sciences (New York) 1 Turkish Journal of Mathematics 1 Georgian Mathematical Journal 1 Differential Equations and Dynamical Systems 1 Nonlinear Dynamics 1 Positivity 1 Communications of the Korean Mathematical Society 1 Communications in Contemporary Mathematics 1 Kragujevac Journal of Mathematics 1 Portugaliae Mathematica. Nova Série 1 Nonlinear Dynamics and Systems Theory 1 Communications on Pure and Applied Analysis 1 Central European Journal of Mathematics 1 Journal of Cosmology and Astroparticle Physics 1 Journal of the Indonesian Mathematical Society 1 Sibirskie Èlektronnye Matematicheskie Izvestiya 1 Nonlinear Oscillations 1 Discrete and Continuous Dynamical Systems. Series S 1 Asian-European Journal of Mathematics ...and 20 more Serials
all top 5

Cited in 28 Fields

 266 Ordinary differential equations (34-XX) 132 Partial differential equations (35-XX) 112 Difference and functional equations (39-XX) 86 Operator theory (47-XX) 77 Global analysis, analysis on manifolds (58-XX) 25 Calculus of variations and optimal control; optimization (49-XX) 25 Mechanics of deformable solids (74-XX) 23 Dynamical systems and ergodic theory (37-XX) 16 Real functions (26-XX) 13 Numerical analysis (65-XX) 12 Functional analysis (46-XX) 7 Integral equations (45-XX) 4 Fluid mechanics (76-XX) 2 Special functions (33-XX) 2 Mechanics of particles and systems (70-XX) 2 Systems theory; control (93-XX) 1 History and biography (01-XX) 1 Mathematical logic and foundations (03-XX) 1 Combinatorics (05-XX) 1 Measure and integration (28-XX) 1 Functions of a complex variable (30-XX) 1 Potential theory (31-XX) 1 Integral transforms, operational calculus (44-XX) 1 General topology (54-XX) 1 Optics, electromagnetic theory (78-XX) 1 Relativity and gravitational theory (83-XX) 1 Astronomy and astrophysics (85-XX) 1 Biology and other natural sciences (92-XX)


text:
We’re rewarding the question askers & reputations are being recalculated! Read more.

# Tag Info

21

In general, all Krylov methods essentially seek a polynomial that is small when evaluated on the spectrum of the matrix. In particular, the $n$th residual of a Krylov method (with zero initial guess) can be written in the form $$r_n = P_n (A) b$$ where $P_n$ is some monic polynomial of degree $n$ . If $A$ is diagonalizable, with $A=V\Lambda V^{-1}$, we ...

17

On norms As an addendum to Reid.Atcheson's answer, I would like to clarify some issues regarding norms. At the $n^{\mathrm{th}}$ iteration, GMRES finds the polynomial $P_n$ that minimizes the $2$-norm of the residual $$r_n = A x_n - b = \big(P_n(A) - 1 \big)b - b = P_n(A) b .$$ Suppose $A$ is SPD, so $A$ induces a norm and so does $A^{-1}$. Then $$\begin{... 15 The matrix you're referring to is positive definite. The eigenvalues of the matrix must be real, because symetric matrices are equal to their own conjugate transpose, and are thus Hermitian. All eigenvalues of Hermitian matrices are real. If all entries of the matrix are positive or zero and the matrix is weakly diagonally dominant, then all eigenvalues of ... 15 There are two main classes of solutions to be discussed in this regard. "Sufficiently" Smooth Solutions In Strang's classical paper it is shown that the Lax equivalence theorem (i.e., the idea that consistency plus stability implies convergence) extends to nonlinear PDE solutions if they have a certain number of continuous derivatives. Note that that ... 11 Iterative methods in a nutshell: Stationary methods are in essence fixed point iterations: To solve Ax=b, you pick an invertible matrix C and find a fixed point of$$ x = x + Cb- CAx $$This converges by Banach's fixed point theorem if \|I-CA\|<1. The various methods then correspond to a specific choice of C (e.g., for Jacobi iteration, C=D^{-1}... 11 In addition to Christian's answer, it's also worth noting that for linear convergence you have e_{k+1} \le \lambda_1 e_k where you have \lambda_1<1 if the method converges. On the other hand, for quadratic convergence you have e_{k+1} \le \lambda_2 e_k^2 and the fact that a method converges does not necessarily imply that \lambda_2 must be smaller ... 10 It isn't so much that we want to compare the p-refinement and h-refinement errors directly, instead we want to compare the convergence properties (e.g. speed) of each refinement strategy. This requires more knowledge of the constant in the apriori error estimate. We'll illustrate by looking at the apriori error estimate of a discontinuous Galerkin ... 10 It is because we typically neglect higher order terms in error estimates. For example, we can show that$$ \|e\| \le C(u) h^2 + {\cal O}(h^3). $$The point is that when h is small, the cubic term is small and can be neglected. In fact, when h is small, you can observe quadratic convergence. But whenever h is not small (where "small" is relative to ... 10 Yes, this is the standard Aubin-Nitsche (or duality) trick. The idea is to use the fact that L^2 is its own dual space to write the L^2-norm as an operator norm$$\|u\|_{L^2} = \sup_{\phi\in L^2\setminus\{0\}} \frac{(u,\phi)}{\|\phi\|_{L^2}}.$$We thus have to estimate (u-u_h,\phi) for arbitrary \phi\in L^2. To do that, we "lift" u-u_h to H^1_0 ... 10 Jean-Michel Muller, et. al., "Handbook of Floating-Point Arithmetic 2nd ed.", Birkhäuser 2018, gives the following example due to Muller, specifically constructed to deliver incorrect results with floating-point evaluation:$$ {u_{0} = 2,\\ u_{1} = -4,\\ u_{n} = 111 - \frac{1130}{u_{n-1}} + \frac{3000}{u_{n-1}u_{n-2}},\>\>\>\>n \ge 2.} $$... 9 In practice, yes. While e_k is still large, the rate coefficient \lambda will dominate the error rather than the q-rate. (Note that these are asymptotic rates, so the statements you linked to only hold for the limit as k\to\infty.) For example, for first order methods in optimization you often observe an initially fast decrease in error, which then ... 9 For a single rational equation in the complex domain, the basin of attraction is fractal, the compelement of a so-called Julia set. http://en.wikipedia.org/wiki/Julia_set . For theory with some nice online figures, see, e.g., http://mathlab.mathlab.sunysb.edu/~scott/Papers/Newton/Published.pdf http://hera.ugr.es/doi/15019160.pdf Even the ''globalized'' ... 9 Let me first answer all the questions: What is the theoretical convergence rate for an FFT Poison solver? The theoretical convergence is exponential as long as the solution is sufficiently smooth. How fast should this energy converge? The Hartree energy E_H should converge exponentially for a sufficiently smooth solution. If the solution is less ... 9 \|x^{(k)}-x^*\| is the error in the kth term, call it E_k. For a "good" numerical method, we want the approximation to get closer and closer to the desired result so E_k has to decrease to zero. If the error is guaranteed to reduce to at least a certain fraction L of the previous step, you have linear convergence:$$E_{k+1} \le L E_k.$$This ... 8 The conjugate gradient algorithm works for semidefinite problems and produces the minimal norm solution. 8 Due to the under-resolved boundary layer near the lid, the velocity in the adjacent cells is significantly lower than the lid. This section is showing you a trick to make the code run faster while still being stable. Increasing the Courant number would normally make the method unstable, but since the velocity in all interior cells is significantly less than ... 8 Let's examine the one-dimensional three-point stencil case in detail, because I think it's important to be clear just how this behaviour arises, and what it means to set a point to a certain value in a finite-difference grid when the underlying function is discontinuous. The equation will be$$ u''(x) = \rho(x). $$Instead of using the interval [-1,1] with ... 8 Yes: See Higham's book "Accuracy And Stability of Numerical Algorithms", second edition, chapter 25: Nonlinear Systems and Newton's Method. In particular, see the section on the "limiting residual" in terms of a condition number for the Jacobian. It may well be that since your system is ill-conditioned, that you quickly hit the limiting residual and your ... 7 The Hartree-Fock equations are the result of performing constrained Newton-Raphson minimization of the energy with respect to the parameter space of Slater determinants (I don't have my copy of Szabo-Ostlund at hand, but I believe this is pointed out in the derivation). Hence, HF-SCF will converge if your starting guess is in a convex region around a minimum.... 7 One cannot conclude from a residual how accurate the solution is. Between the best and the worst case in norm, there is a factor of exactly the condition number. More precisely, if the residual norm is r and the error norm is e then \|A\|^{-1}\le e/r \le \|A^{-1}\|, and both bounds are attainable. Taking the quotient of the bounds proves the claim. The ... 7 Convergence is a statement about the asymptotic limit h\to0. Therefore, it's not possible to "prove" convergence by any finite number of computations. The best you can do with computations is motivate the belief that an algorithm does converge. To prove convergence, you need to...write a proof. 7 If analytic techniques are disallowed but the periodic structure is known, here is one approach. Let$$g(x) = \frac{\cos x}{2-\cos x}$$be periodic with period 2 \pi, so that$$g(x) = \sum_j w_j e^{ijx}$$where$$w_j = \frac{1}{2\pi}\int_0^{2\pi} g(x) e^{-ijx} dx$$Thus,$$\begin{aligned} f(x) &= \sum_{k \ge 1} \frac{g(kx)}{k^p} \\ &= \sum_{k ...

7

The Taylor-Hood approximation of the Stokes flow is a mixed finite element method, for which error estimates generally have the form $$\|u-u_h\|_V + \|p-p_h\|_M \leq C (\inf_{w_h\in V_h}\|u-w_h\|_V + \inf_{q_h\in M_h}\|p-q_h\|_M), \tag{1}$$ where $(u,p)\in V\times M$ is the exact solution and $(u_h,p_h)\in V_h\times M_h$ is the approximation. For the ...

7

If your matrix is symmetric, positive definite, the CG method may converge slowly, but it converges for $n\to\infty$. The only reason it does not converge on a computer are round-off errors, in particular if the condition number of the matrix, the quotient of largest and smallest eigenvalue is large. Experience is, that with double precision arithmetic, ...

7

The two normalization formulas result in two different algorithms, that they both "normalize" a vector is not so relevant. As an example, consider the following transition matrix: $$M = \begin{pmatrix}0&1\\1&0\end{pmatrix}.$$ Starting with $r=(1,0)$, consecutive normalized vectors $r$ will be $(0,1)$ and $(1,0)$, so the algorithm clearly will not ...

7

The issues you're running into now are not a failing of Newton-Raphson, but a question of coupling. You're doing iterated sequential coupling -- solving each equation sequentially and then iterating until (hopeful) convergence. No solver choice in place of NR is going to fix this lack of convergence, as long as you are doing iterated sequential coupling. ...

7

Standard terminology in nonlinear (i.e., derivative-based) optimization is that "global convergence" means "convergence to a stationary point no matter where you start from" (where the limit may in fact depend on the starting point). This is in contrast to local convergence, which requires that you start sufficiently close to a stationary point; if your ...

7

Unfortunately, convergence of GMRES does not have a clear dependence on the distribution of eigenvalues. It was proved by Greenbaum, Ptak and Strakos in 1996 that you can construct examples with an arbitrary spectrum and an arbitrary convergence history: that is, give me any $n$ nonzero complex numbers, and any decreasing sequence $\|r_k\|$, and I can ...

6

The interpretation is qualitatively correct. Note that linear and quadratic convergence are with respect to the worst case, the situation in a particular algorithm can be better than what you get from the worst case analysis given by Wolfgang Bangerth, though the qualitative situation usually corresponds to this analysis. In concrete algorithms (e.g., in ...

6

For many partial differential equations arising in nature, particularly with strong nonlinearities or anisotropies, the choice of an appropriate preconditioner can have a large effect on whether the iterative method converges rapidly, slowly, or not at all. Examples of problems that are known to have fast and effective preconditioners include strongly ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Experimental Analysis with Several Discrete Treatments

I am analyzing data that are originating from an randomized experiment and I am new to this. There was one control groups and three different treatment groups. The treatment groups are discrete and unordered (you can think of them as of three completely different drugs).

I want to analyze the data employing an regression.

At first I subseted the data (so that only one treatment group and the control group were left) and analyzed the following:

$$y_i=\alpha+\beta T_i + error_i$$

where $$T_i$$ is a dummy indicating whether an individual received the treatment. In this case, $$\beta$$ corresponds to the average causal effect of the treatment.

I have a question: Would, instead of subsetting, also the following expression feasible?

$$y_i=\alpha + \beta_1 T_{1,i} + \beta_2 T_{2,i} + \beta_3 T_{3,i} + error_i$$

where $$T_{1,i}$$, $$T_{2,i}$$, $$T_{3,i}$$ are dummies indicating whether an individual received the first, second or third treatment.

Best regards

• Yes, the model with 3 dummy variables is correct. – user158565 Nov 5 '18 at 14:36
• However, the dummies are correlated right? This might lead to bigger standard errors? However, due to the greater subset, the estimate might get more precise. I don't know which of these effects might be stronger :O – FeldO Nov 6 '18 at 13:08
• You have 4 groups (one control groups and three different treatment groups.), so 3 dummy variables will not correct collinearity (correlation). If you create 4 dummy variables, collinearity will appear. After you fit the model, check the 3 estimated 3 regression coefficients to see which drug is the best. – user158565 Nov 6 '18 at 18:44

If you are determined to use regression, you should include all the treatment dummies and all the data points. The reason for doing this is to increase the precision of your estimate of the mean squared error, which is part of the test statistic for each group comparison. One problem with regression is that the dummy codes only provide tests of each treatment against the reference (i.e., control) group. If you want to make other comparisons, you need to perform additional Wald tests for the differences in coefficients (this is straightforward in Stata using test). If you perform many Wald tests, your Type I error rate increases unless you correct for it. Many ANOVA procedures have this built in, which is why I recommend it over regression unless your only question is whether each treatment group differs from control.


text:
# There is a natural morphism $\lim\limits_{\longleftarrow} \beta \to \lim\limits_{\longleftarrow} \beta \circ \varphi^{op}$?

This is from Categories & Sheaves by Kashiwara & Schapira.

$$\text{Hom}_{\text{Set}}(X, \lim_{\leftarrow} \beta) \xrightarrow{\sim} \lim_{\leftarrow} \text{Hom}_{\text{Set}}(X, \beta) \tag{2.1.1}$$

Here's the exercise:

Let $\varphi : J \to I$ and $\beta : I^{op} \to \text{Set}$ be functors. Denote by $\varphi^{op} : J^{op} \to I^{op}$ the associated functor. Using (2.1.1), we get a natural morphism:

$$\lim_{\longleftarrow} \beta \to \lim_{\longleftarrow}\beta \circ\varphi^{op}$$

Firstly, I don't see how the left or right is a functor so how can we speak of the natural map?

Secondly, how do you make the map? If we have a natural map $a \in \lim\limits_{\longleftarrow} \beta$ such that $\beta(f) \circ a_i = a_j$ for any $f : j \to i$ in $I$, I'm not seeing how this should work.

• Surely there must be additional conditions on $\phi$? This would certainly fail if, for example, $J$ was the empty category, since it would imply that every limit is the terminal object, and thus that every set is a singleton. – Arnaud D. Jun 12 '18 at 14:36
• Which exercise is this? Regarding the issue of how both sides are functors, in the book they also refer to the functor $X\mapsto \lim \mathrm{hom}(X,\beta)$ as $\lim \beta$, which in the case the limit exists in the respective category is the Yoneda embedding of the limit. – asdq Jun 12 '18 at 14:51
• It's on page 36. – StudySmarterNotHarder Jun 12 '18 at 14:53
• On page 36, I read a natural morphism (not isomorphism). If $C$ has all $I$-limits, you can make $\varprojlim\beta$ a functor of $\beta$, in other words $\varprojlim:\operatorname{Fun}(I,C)\to C$. Similarly if $C$ has all $J$ limits, you have a functor $\operatorname{Fun}(I,C)\to C$ such that $\beta\mapsto\varprojlim\beta\circ\varphi^{op}$. The claim is that this is a natural transformation. – Roland Jun 12 '18 at 17:12
• @Roland I see that part now, thank you. How do I show that there is a map between the two though? – StudySmarterNotHarder Jun 12 '18 at 17:49

Let's dispense with all the duals and the values taken in a particular category, which are obscuring the situation a bit. If $D:J\to C$, $u:I\to J$ are functors such that $C$ admits limits of shapes $I$ and $J$, the claim is that we have a natural map $\mathrm{lim} D\to \mathrm{lim}(D\circ u)$. For any $i\in I$, we must give a map $\mathrm{lim} D\to D(u(i))$.
Now, along with $\mathrm{lim} D$ we are given a cone $\lambda$ with components $\lambda_j:\mathrm{lim} D\to D(j)$. So the natural guess for the desired map is $\lambda_{u(i)}$. To prove that these components determine a map into $\mathrm{lim}(D\circ u)$ as desired, we have only to check that $(D\circ u)(f)\circ \lambda_{u(i)}=\lambda_{u(i')}$ for every $f:i\to i'$ in $I$. But this is simply an instantiation of the assumption that $\lambda$ forms a cone.
This proof uses just the definition of $$\lim\limits_{\leftarrow} \beta = \text{Hom}_{I^{\wedge}}(\text{pt}, \beta)$$ and the fact that $$\beta(f) \circ \theta_y = \theta_x$$ for all natural maps $$\theta$$ in the limit, and $$f : x \to y$$ in $$I$$.
Suppose also that since $$\varphi(g) \in I^{op}, \forall \ g \in J^{op}$$, we have $$\varphi(g): \varphi(b) \to \varphi(a)$$ for all $$g \in J^{op}$$ so that by letting $$f = \varphi(g)$$ in the above we get that $$\beta(\varphi(g)) \circ \theta_{\varphi(b)} = \theta_{\varphi(a)}$$. Thus for any $$\theta$$ in the first limit, there is $$\theta \circ \varphi \equiv \theta_{\varphi(\cdot)}$$ which is a natural map in $$\text{Hom}_{J^{\wedge}}(\text{pt}, \beta \circ \varphi)$$.


text:
# B How can log(x) = ln(x)/ln(10)?

1. Aug 7, 2016

### Natalinatul

Might be a silly question but I'm trying to search for ways to prove that it's right without using the numbers itself

2. Aug 7, 2016

### Staff: Mentor

Does the following equation make sense to you:
$$x=10^{log x}$$
It says that, by definition, the log of x is the power to which you have to raise 10 to get x.

3. Aug 7, 2016

Chestermiller's answer is correct, and so by this time you may have figured this out, but if you are still having difficulty, just google the proof of the change of base formula. Or just google the change of base formula, and many sites will also offer the proof (based on Chestermiller's hint)

4. Aug 7, 2016

### Natalinatul

Yup.. How could I miss that! The best thing about his answer was the fact he didn't answer it but gave a great hint... I figured it out now...

5. Aug 9, 2016

### Ssnow

Hi, there is this formula $\log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}}$ that permits to change the base from $b$ to $c$.

6. Aug 9, 2016

### Staff: Mentor

Don't you mean when you change the base from c to a?

7. Aug 9, 2016


text:
# Tag Info

0

The definition of rank is the highest order of a square non degenerate submatrix. Thus being determinant invariant under transposition, you'll always have $\operatorname{rk}A=\operatorname{rk}A^T$ for every $A\in M_{n,m}(R)$.

1

The two following properties are easy set-theoretic statements $Im(f\oplus g)= Im(f)\oplus Im(g)$ $Ker(f\oplus g)=Ker(f)\oplus Ker(g)$ Those two results give the equivalence you want.

4

Consider $R=\mathbb Z$, $M=\mathbb Z$ and $N=\mathbb Z/2\mathbb Z$.

1

Hint: Prove the statements below ( may assume $I \ne 0$ ): If $I$, $J$ are ideals of an integral domain $R$ with field of fractions $K$ then every morphism of $R$-modules $I \to J$ is given by the multiplication by an element in $K$. If $I$ is a projective module then there exists an imbedding of $I$ into a free $R$ module $i \colon I \hookrightarrow$ ...

2

This result is known as Schanuel's lemma. A quick proof is given by introducing the pullback of $f$ and $f'$. It is the submodule of $P\oplus P'$ given by $$X = \{(p,p')\in P\oplus P'\mid f(p) = f(p')\}.$$ Then, the following sequences are exact : $$0 \longrightarrow \ker(f')\simeq K' \longrightarrow X \longrightarrow P \longrightarrow 0$$ and $$0 ... 0 Assume a \in IQ and a \not\in Q and then work from the element in I which is not a zero divisor. 0 As it has been mentioned in the above comments, M\otimes_SN is an R-T bimodule with the following actions; let r\in R , t\in T and m\otimes n\in M\otimes_SN ( a pure tensor) we have$$r.(m\otimes n)=r.m\otimes n$$and$$(m\otimes n).t=m\otimes(n.t)$$these actions are also compatible, since r.((m\otimes ... 1 It is clear that the product is injective wrt itself and from a theorem that a product is injective iff each factor is injective the first part is settled. As for the second part we need to see that we cannot inject the product inside the direct sum since we have infinitely many factors, see Ribenbenboim : Rings and modules top of page 25. 1 The inclusion \bigoplus R_n \hookrightarrow \prod R_n is essential, hence \bigoplus R_n is not injective. 1 One should approch this more conceptually: The uniqueness is an immediate consequence of the surjectivity of f_2 (You can view this as a definition of the term surjective). The existence follows from the universal property of the cokernel, which is the same as the fundamental homomorphism theorem: The morphism g_2 \circ \beta annihilates anything, ... 2 I think you want I to be a right ideal and J a left ideal, not the other way round. You are almost done. Consider the canonical homomorphism \mathbf{m} :A\otimes_{A}A\rightarrow A of \mathbb{Z}-modules which sends every a\otimes b\in A\otimes_{A}A to ab\in A. It is well-known that \mathbf{m} is an isomorphism. Thus, \left( ... 1 You have confused k-vector space basis as R-module basis, hence this issue. ANy commutative ring is a free module overitself with basis the singleton \{1\}, consiting of the multiplicative identity. Given two elements a,b\in R, by commutativity we have a.b -b.a=0 and so a basis has to have less than 2 elements. 1 As you remarked, there are four non-trivial subgroups of \mathbb Z_{18} given by the divisors of 18 (others than 1 and 18). Now it's good to know what's the sum and the intersection of such subgroups. In general, in \mathbb Z_n we have \bar k\mathbb Z_n+\bar l\mathbb Z_n=\overline{\gcd(k,l)}\mathbb Z_n and \bar k\mathbb Z_n\cap\bar l\mathbb ... 0 This holds whenever R is noetherian; see Bourbaki, Commutative Algebra, Chapter 4, Exercise 17(g). (In fact, it's enough to know that P is minimal over \operatorname{Ann}(m) for some m\in M, m\ne 0.) But this isn't the case in your question. However, it also holds under the assumption M noetherian. In this case the ring R/\operatorname{Ann}(M) ... 1 To augment user26857's answer, note that: (a_1,a_2,\dots,a_n) + S = (a_1+a_2+\cdots+a_n,0,\dots,0) + (a_1-a_2-\cdots-a_n,a_2,\dots,a_n) + S = (a_1+a_2+\cdots+a_n,0,\dots,0) + S. If p(X) = q(X)(X - 1) + r (for r \in A), then p(X) + S = r + S. If B = \begin{bmatrix} x_{11} & x_{12} & x_{13} & \dots & x_{1n} \\ ... 1 If P is a finitely generated and projective R-module, then there is a canonical isomorphism$$\zeta:P^*\otimes_RM\simeq\operatorname{Hom}_R(P,M)$$given by \zeta(f\otimes m)(x)=f(x)m. (For a proof see Proposition 6 from these notes.) Now, coming back to the question we have \mathfrak a^*\otimes_RM\simeq\operatorname{Hom}_R(\mathfrak a,M). But ... 1 Let b_1+A,b_2+A,\dots,b_m+A be generators of B/A and let a_1,\dots,a_n be generators of A. If b\in B, then$$ b+A=\sum_{i=1}^m r_i(b_i+A)=\biggl(\sum_{i=1}^m r_ib_i\biggr)+A $$for some r_1,\dots,r_m\in R, which means$$ b-\biggl(\sum_{i=1}^m r_ib_i\biggr)=\sum_{j=1}^n s_ja_j $$for some s_1,\dots,s_n\in R, so$$ b=\sum_{i=1}^m ...

2

The problem asks to find an $A$-module $N$ (eventually you already know from class) such that $M/S\simeq N$ (eventually by using a well known isomorphism theorem). Hints. Define $f:A^n\to A$ by $f(a_1,\dots,a_n)=\sum_{i=1}^na_i$. Define $f:A[X]\to A$ by $f(p(X))=p(1)$. Define $f:M_n(A)\to A^n$ by $f((a_{ij})_{i,j})=(a_{11},\dots,a_{n1})$.

0

Another example is $A=\mathbb C$ and $B=\mathbb R$. One knows that $\mathbb R$ is a $\mathbb C$-vector space, but there is no ring homomorphism $\mathbb C\to\mathbb R$ (why?).

1

This is well-studied under the heading of "cancellability," and Lam's crash course on the topic is very nice. Are there any simple conditions on $R$-modules $M,A$ and $B$... The readiest one is that if $R$ has stable range 1 and $M$ is finitely generated and projective, then it cancels from $M\oplus A\cong M\oplus B$. You can find this, for example, in ...

3

An example of a difference: $\mathbb{Z}/3\mathbb{Z}$ is projective (and free) as a $\mathbb{Z}/3\mathbb{Z}$-module but is not projective (nor free) as a $\mathbb{Z}$-module. However things like simplicity and indecomposability will remain the same, which is easy to prove since the action of $A/\mathfrak a$ is given by the action of $A.$ The only thing ...

4

If you want $\operatorname{Hom}_R(P,R)\simeq P$ for $P$ a finitely generated projective $R$-module, then forget it. $\bullet$ If $R$ is an integral domain, and $I\subset R$ is a non-zero ideal, then $\operatorname{Hom}_R(I,R)\simeq I^{-1}$, where $I^{-1}=\{x\in Q(R):xI\subseteq R\}$. (Here $Q(R)$ stands for the field of fractions of $R$.) $\bullet$ If ...

2

Hint: what exact sequence can you build up out of $M_1\oplus M_2, M_1 + M_2$ and $M_1\cap M_2$?

0

Rephrased, to show that the function is well-defined, you have to show that whenever $g^m|(f_1-f_2)$, you also have $(t-\lambda)^m|(f_1-f_2)$. But this is clear since $(t-\lambda)^m|g^m$.

1

$\mathbb{C}[M]$ will be Laurent polynomials in $x_1,\dots,x_n$. Algebra homomorphisms to $\mathbb{C}$ are thus determined by sending each $x_i$ to an element of $\mathbb{C}^\times$ and therefore the set of all such homomorphisms is associated with $(\mathbb{C}^\times)^n$ More generally, the group algebra functor is left adjoint to the functor taking an ...

0

Yes. If $R$ has infinite left global dimension, then for every $n$ there is a left module $M_n$ with projective dimension greater than $n$. But then $\bigoplus_{n\in\mathbb{N}}M_n$ has infinite projective dimension. The same argument works for weak dimension.

1

In the general case, there is a canonical isomorphism: \begin{align*}P/N\cap P &\longrightarrow(P+N)/N\\ x+N\cap P & \longmapsto x+N\end{align*} Also, there's a bijection between submodules of $M/N$ and submodules of $M$ that contain $N$, and $P$ is not supposed to contain $N$. However, by the canonical homomorphism $\,p\colon M\longrightarrow M/N$, ...

1

From the artinian hypothesis you may deduce that the zero submodule is an intersection of finitely many maximal submodules. Therefore $M$ is isomorphic to a submodule of a finitely generated semisimple module, hence it is semisimple and finitely generated.

1

Let $\mathcal{M}$ be the set of maximal submodules of $M$. Lemma: There exist a finite subset $\mathcal{N}$ of $\mathcal{M}$ such that $\bigcap \mathcal{N} = 0$. Proof: Let $\mathcal{C}$ be the collection of the intersection finite number of maximal submodules. As $M$ is artinian, $\mathcal{C}$ must have a minimal element $N = \bigcap \mathcal{N}$. If it ...

-1

I am not sure I fully understand the question. As Hoot said, "is $R$ artinian?" to which you replied as "yes". But you also have the finitely generated assumption. In which case the assumption that $M$ is semisimple is completely unnecessary. A finitely generated module over an Artinian ring is Artinian. Because, we have the exact sequence, $$0 \to K \to ... 2 Your method is not going to work unless you're careful about the choice of the projective modules P and Q through which gf-\text{id}_M and fg-\text{id}_N factor. Indeed, you could replace P by P\oplus P' for any projective module P', changing the isomorphism type of M\oplus P. Instead, use the fact that gf-\text{id}_M factor through a ... 1 Write M as the direct sum of its irreducible submodules. Each element of the generating set can be written as a sum of elements from finitely many of the irreducible submodules. Then M is a direct sum of finitely many irreducible submodules. This implies M is Artinian. 2 Without any assumptions about R, this is false. For example, consider \mathbb{Z} as a \mathbb{Z}-module. It's finitely generated (the set \{1\} is a generating set), but the collection of submodules$$\mathbb{Z}\supset 2\mathbb{Z}\supset 4\mathbb{Z}\supset 8\mathbb{Z}\supset\cdots$$has no minimal element. 1 In different categories (such as the category of rings or the category of modules) different substructures are important. Ideals are the natural substructures of rings, and submodules are the natural substructures of modules. These substructures are special because they are suitable for forming quotient objects in their categories (that is why I am not ... 1 When that definition says Let R be a ring and 1_R its multiplicative identity... they are mentioning (implicitly) that they are requiring R to be unital. However, it is certainly possible to define modules over a non-unital ring; just throw out statement 4 from that definition. 1 Some authors use the term "ring" to mean "ring with identity." This is neither standard nor nonstandard; there is just no consensus. Authors using this definition would use the term "rng" to denote a ring that possibly does not have an identity. 1 I don't have the book by Dummit and Foote, but the maps$$ R\to Ry_1,\quad r\mapsto ry_1\\ R\to Ra_1y_1,\quad r\mapsto ra_1y_1 $$are surjective homomorphisms and they're also injective because M is torsion-free. For the second one, note that ra_1=0 implies r=0, if a_1\ne0. If a_1=0, then Ra_1y_1=\{0\} is free as well. 0 For a module such as B, with submodules C and D, we of course have the submodule C + D of B, which is, by definition, C + D= \{ c + d \mid c \in C, d \in D \} \subset B; \tag{1} in the event that C \cap D = \{ 0 \}, the notation C \oplus D may also be used; C \oplus D is then referred to as the direct sum of C and D. We observe that ... 2 Embed L in an injective module E and consider the push-out diagram$$\require{AMScd} \begin{CD} {} @. {} @. 0 @. 0 \\ @. @. @VVV @VVV \\ 0 @>>> K @>>> L @>>> M @>>> 0 \\ @. @| @VVV @VVV \\ 0 @>>> K @>>> E @>>> N @>>> 0 \\ @. @. @VVV @VVV \\ {} @. {} @. E/L @= E/L \\ @. @. @VVV @VVV ...

2

For a $b\in B$ consider the element $x=f (g(b)) -b\in B$. We claim that this $x$ is in the kernel of $g$. Then it will follow that $b= x+f(g(b))$ where in the rhs first element is in $\ker g$ and the second element is in image of $f$, that is what you wanted to be proved. To prove the claim apply $f$ to this element $x$ and use the condition that $g$ ...

1

It seems like your question is: prove that a finitely generated free module has a finite basis. Let $E$ be a basis for such a module $M$ over $R$. Then $E$ spans $M$. Now, $M$ is finitely generated so you can find a finite ste of elements that span $M$. Write each element as (finite) linear combinations of elements from $E$. Collect all these elements from ...

4

$\mathbb{Z}/6\mathbb{Z}$ is not a PID, because it's not a domain.

1

I am surprised that it is not mentioned here- Example of a free module M which has bases having different cardinalities. Let $V$ be a vector space of countably infinite dimension over a division ring $D$. Let $R=End_D(V)$. We know that $R$ is free over $R$ with basis $\{1\}$. We claim that given a positive integer $n$, there is a $R$-basis $B_n=\{f_1,f_2, ... 2 We want to show${\rm Ext}^1( P, N ) = 0$for all$N$(as this is equivalent to$P$being projective). Accordingly, let $$0 \to N \to I^0 \mathop\to^{\delta^0} I^1 \mathop\to^{\delta^1} I^2 \to \cdots$$ be the beginning of an injective resolution for$N$. Suppose$f\colon P \to I^1$is a cocycle, i.e.,$\delta^1\circ f = 0$. Since the sequence is exact, ... 1 You can invert$r$as long as$r$is not nilpotent (otherwise the multiplicative set generated by$r$contains$0$and the localizations are also$0$). Since$r$is a non-zero divisor on$L$it follows that$r$is not nilpotent (unless$L=0$which I think it's not the case due to the rank condition). 1 Since$\Bbb Z$is a PID and projective modules over PIDs are free, it is clear that every finite abelian group is not projective. But in fact, you can just use the definition to prove that$\Bbb Z/2\Bbb Z$is not projective as$\Bbb Z$-module: consider the quotient map $$\pi:\Bbb Z/4\Bbb Z\longrightarrow\Bbb Z/2\Bbb Z.$$ It is easy to check that there is ... 0 Any projective module over a PID (e.g.,$\mathbb{Z}$) must be free. 2$\mathbb Z/2\mathbb Z{}{}{}{}{}{}{}{}$. 1 Your proof is incorrect, sorry. The module$M/S$is not a submodule of$M$and indeed need not be torsion-free. Example. Consider$R=\mathbb{Z}$; then$\mathbb{Z}=\langle 2,3\rangle$, but$\mathbb{Z}/\langle 3\rangle\$ is not torsion-free.

2

The usual proof uses the Structure theorem for finitely generated modules over a principal ideal domain. But there is an alternative proof based on the fact that the finitely generated torsion-free modules over an integral domain are isomorphic to a submodule of a free module of finite rank. (For a proof see here.) Since over a PID the submodules of free ...

Top 50 recent answers are included


text:
Comment
Share
Q)

# A paisa coin is made up of Al-my alloy and weights $0.75\;g$ It has a square shape and its diagonal measures $17\;mm$ It is electrically neutral and contains equal amounts of positive and negative charges. Treating the paisa coins made up only $Al$, find the magnitude of equal number of positive and negative charges. What conclusion do you draw from this magnitude ?

Comment
A)
Let us consider that the coin is made of Aluminium
Given , weight of the diagonal $=17 \;mm$
Atomic mass of Al = 27 g
Atomic number of Al =13
One molar mass of substance has $6.023 \times 10^{23}$ number of atoms
$27\;g$ Al contains $6.023 \times 10^{23}$ atom
1g of Al contains $=\large\frac{6.023 \times 10^{23}}{27}$ atom
$0.75g$ of Al contains $\large\frac{6.023 \times 10^{23}}{27}$$\times 0.75$
$\qquad= 1.67 \times 10^{23}$
One atom of Al contains 13 electrons 13 protons
$1.67 \times 10^{22}$ atom of Al contains
$13 \times 1.67 \times 10^{22}= 21.71 \times 10^{22}$ electron
Hence number of proton $=21.71 \times 10^{22}$
The charge on one electron $= 1.6 \times 10^{-19}$
Charge on $21.71 \times 10^{22}$ electrons.
$\qquad= 1.6 \times 10^{-19} \times 21.17 \times 10^{22}$
$\qquad= 34.736 \times 10^3 \;C$
Thus the mount of positive and negative charge on the coin $=34.736 \times 10^{3}\;C$
This is a very large amount of charge .
This concludes that all the atoms contain a very large amount of positive and negative charges.


text:
Precision of Success

I have a setup where I am making X number of measurements over a particular time frame. These X measurements are success/failure measurements. For example, in the period of an hour I might get 422 successes and 584 failures. Then I calculate the success rate for that hour of measurements being 0.72260273972. Up to what decimal place can I be sure that I am calculating a meaningful number? Is it a function of the number of measurements in that hour or simply the significant figures used in the calculation.

Some help would be much appreciated!

This is the oft-encountered problem of estimating a "confidence interval for a Binomial proportion". There are several different techniques, which vary in their assumptions and are well described on the Wikipedia page.

For all reasonable approaches, the precision (i.e. width of the confidence interval) will depend on

1. your sample success ratio (i.e. $\hat{p}=\frac{422}{n}$)
2. the number of measurements (i.e. $n=422+584$)
3. the desired "confidence level" (i.e. a number $\alpha\in(0,1)$ giving the fractional coverage of the associated interval estimate)

(Note: I put some terms in quotes to try and steer clear of the thorny issues surrounding confidence-intervals vs. credible-intervals, $p$-values, etc.)

Update: At the request of the OP, here is some clarification on how the interval estimate would relate to the numerical precision.

For simplicity, I will assume that "$d$ significant digits" can be interpreted as "$\pm\,5\times 10^{-(d+1)}$", i.e. $$\hat{p} = 0.722 \implies p = \hat{p} \pm 0.0005 \implies 0.7215 < p < 0.7225$$ where $\hat{p}$ is the estimate of the true value $p$. So a significance of $d$ digits roughly corresponds to an interval width of $10^{-d}$.

Again for simplicity, assume we use the "normal approximation", so for a given "confidence level" (specified as a "$z$-score") we have interval width $$\frac{\Delta{p}}{z}=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$ So, for example if we use a coverage probability of 95%, corresponding to $z=1.96$ (roughly 2 standard deviations), then for $\hat{p}=\frac{422}{n}$ and $n=422+584$, we have $$\Delta{p}\approx 3 \times 10^{-2} = \mathrm{O}[10^{-2}]$$ so, keeping things simple, we might say that there are roughly two significant digits. So while nominally we have $\hat{p}\approx 0.419483101391650$, from these calculations $p\approx 0.42$ would be a reasonable inference (with the $\pm 0.03/2$ implicit in the significant-digit truncation).

• How does this allow me to know whether 0.72260273972 is a meaningful number or nonsense as opposed to 0.722, where 8 digits were truncated? – user2253546 Oct 26 '16 at 2:57
• @user2253546 does the update clarify things? – GeoMatt22 Oct 26 '16 at 3:49
• Yes it definitely does. I can see the reasoning behind it. Would you have sources of where this is used? Btw, thank you for the well explained response :) – user2253546 Oct 26 '16 at 4:59
• I pieced together this approach from the "standard ingredients" in the Wikipedia links contained in my answer (most of those pages have sources, but no aggregated ref I can give). In terms of "significant digits", this post presents some reasonable guidelines, and this one gives a possible general reference* that is from a pretty official source (*i.e. it probably does not present the particular method I presented here). – GeoMatt22 Oct 26 '16 at 5:26
• GUM reference online here. I cannot find a good cite for it on Google Scholar, but this appears similar, and may be the predecessor (NIST/USA vs. BIPM/France). – GeoMatt22 Oct 26 '16 at 5:37


text:
New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education

The Reductive Subgroups of $$F_4$$
David I. Stewart, New College, Oxford, United Kingdom
 SEARCH THIS BOOK:
Memoirs of the American Mathematical Society
2013; 88 pp; softcover
Volume: 223
ISBN-10: 0-8218-8332-1
ISBN-13: 978-0-8218-8332-7
List Price: US$69 Individual Members: US$41.40
Institutional Members: US\$55.20
Order Code: MEMO/223/1049

Let $$G=G(K)$$ be a simple algebraic group defined over an algebraically closed field $$K$$ of characteristic $$p\geq 0$$. A subgroup $$X$$ of $$G$$ is said to be $$G$$-completely reducible if, whenever it is contained in a parabolic subgroup of $$G$$, it is contained in a Levi subgroup of that parabolic. A subgroup $$X$$ of $$G$$ is said to be $$G$$-irreducible if $$X$$ is in no proper parabolic subgroup of $$G$$; and $$G$$-reducible if it is in some proper parabolic of $$G$$. In this paper, the author considers the case that $$G=F_4(K)$$.

The author finds all conjugacy classes of closed, connected, semisimple $$G$$-reducible subgroups $$X$$ of $$G$$. Thus he also finds all non-$$G$$-completely reducible closed, connected, semisimple subgroups of $$G$$. When $$X$$ is closed, connected and simple of rank at least two, he finds all conjugacy classes of $$G$$-irreducible subgroups $$X$$ of $$G$$. Together with the work of Amende classifying irreducible subgroups of type $$A_1$$ this gives a complete classification of the simple subgroups of $$G$$.

The author also uses this classification to find all subgroups of $$G=F_4$$ which are generated by short root elements of $$G$$, by utilising and extending the results of Liebeck and Seitz.

• Reductive subgroups of $$F_4$$


text:
# Math Help - Complex (non-real) factors of a polynomial

1. ## Complex (non-real) factors of a polynomial

Hi the question says If z-3i is a factor of 2z^4-4z^3+21z^2-36z+27

I tried doing long division by dividing it by z-3i but i get a perfect zero as a answer so i cant do much with that, is their another way i can approach this.? and im working in the chapter factorasation of polynomials of C.

thanks

2. Complex roots come in conjugate pairs. Thus, since $3i$ is a root, so is $-3i$. Thus $z-3$ and $z+3$ are factors. So the polynomial is divisible by $(z-3i)(z+3i)=z^2+9$.

You can now perform long division to get a quadratic polynomial. You can then find the roots of this quadratic polynomial by completing the square or using the quadratic formula.

Remark: Instead of doing long division with $z^2+9$, you can perform synthetic division twice with $3i$ and $-3i$.

3. Since all the coefficients of your polynomial are real, you know that complex roots occur as conjugates.

Since $\displaystyle z - 3i$ is a factor, so is $\displaystyle z + 3i$, which means $\displaystyle (z - 3i)(z + 3i) = z^2 + 9$ is a factor.

So divide your polynomial by $\displaystyle z^2 + 9$ to get the remaining quadratic factor, which you can then factorise further into two linear factors.

4. Hello iFuuZe,

If you mean that you get zero as the remainder, that's good.


text:
Optimization Help!

So I have a problem. The problem says:
A fence 8 ft tall rubs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

I drew a picture to help, but I can't draw it on here. I drew a right triangle, it's fairly large. Halfway through the base, there's a straight line up representing the fence. I then made the hypotenuse of the entire triangle L. The wall is h.

1.I then set up a proportion. It's (x/8)=((x+4)/(h))
2.I solved the proportion for x and got 32/(h-8)
3.I then said (x+4)^2+h^2=L^2
I plugged in the x value i solved the proportion for into the above equation and got
(((4h)/(h-8))^2)+(h^2)^(1/2))=L (It might help to right this out instead of looking at it.

Now I have to take the derivative of this and set it equal to zero.. But I keep messing it up.. Can somebody show me how to do it?

I have been working and realized I can take the derivative of the Pythagorean theorem mentioned in step 3. But I still have trouble solving this. can somebody please help?

Last edited:

Related Calculus and Beyond Homework Help News on Phys.org
hotvette
Homework Helper
This problem is a lot easier if you use trig functions. Start with 4+x = L*cos(θ) and write an expression for x in terms of θ using the small triangle. Substitute and differentiate L with respect to θ. It works out quite nicely.

dynamicsolo
Homework Helper
1.I then set up a proportion. It's (x/8)=((x+4)/(h))
2.I solved the proportion for x and got 32/(h-8)
3.I then said (x+4)^2+h^2=L^2
I plugged in the x value i solved the proportion for into the above equation and got
(((4h)/(h-8))^2)+(h^2)^(1/2))=L (It might help to right this out instead of looking at it.

Now I have to take the derivative of this and set it equal to zero.. But I keep messing it up.. Can somebody show me how to do it?
I just did this problem with a student this past week (Stewart, Section 4.7, no?). The trick if you do this algebraically is not to simplify your expression too soon. It will also be better if you eliminate h , rather than x . Your Pythagorean result becomes

$$L^{2} = (x + 4)^{2} + (\frac{8 \cdot (x+4)}{x})^{2} .$$

There are two things that will be helpful to do. One is to factor out the ( x + 4 )2. The other is to differentiate the expression for L2 implicitly with respect to x . When you are looking to minimize a length given by the Pythagorean Theorem or the distance formula, since that length is a positive number, minimizing L2 also minimizes L . You avoid dealing with a radical until you absolutely have to (if you need to at all).

We will then have

$$\frac{d}{dx} L^{2} = \frac{d}{dx} [ (x + 4)^{2} [ 1 + \frac{64}{x^{2}}] ] = 0$$

$$\Rightarrow 2L \cdot \frac{dL}{dx} = 2 \cdot (x + 4) ( 1 + \frac{64}{x^{2}}) + [ (x + 4)^{2} ( - \frac{128}{x^{3}}) ] = 0$$

Since L is not equal to zero , we can just focus on getting the right-hand side of this equation to be zero. Also, x ≠ -4 , so we can divide that factor out of the equation; we can also take out a factor of 2 . This will leave

$$( 1 + \frac{64}{x^{2}}) + [ (x + 4) ( - \frac{64}{x^{3}}) ] = 0 \Rightarrow 1 + \frac{64}{x^{2}} = (x + 4) ( \frac{64}{x^{3}}) \Rightarrow 1 + \frac{64}{x^{2}} = \frac{64}{x^{2}} + \frac{256}{x^{3}} \Rightarrow \frac{256}{x^{3}} = 1 .$$

You can take it from there (I'll tell you that the minimized length is ≈16.65 ft.)

The trigonometric method hotvette describes is the one the solver for Stewart's solution manual used. It is "easier" in the sense that the differentiation to find the critical value for $\theta$ is simpler; you pay for it later in extracting the minimal length for the ladder (particularly if you are solving for the "exact" value). You get a different but equivalent expression for the length, which is confirmed when you calculate the decimal approximation.


text:
Suffix trees are an application to particularly fast implement many important string operations like searching for a pattern or finding the longest common substring.

## Introduction and definitions

I already introduced the Z-Algorithm which optimizes the searching for a pattern by preprocessing the pattern. It is very useful if you have to search for one single pattern in a large number of words. But often you’ll try to find many patterns in a single text. So the preprocessing of each pattern is ineffective. Suffix trees come up with a preprocessing of the text, to speed up the search for any pattern.

As expected a suffix tree of the word $$S\in\Sigma^+$$ (length $$|S|=m$$) is represented in a data structure of a rooted tree. Every path from root to a leave represents a suffix of $$S\$$$with $$\\notin\Sigma$$. The union $$\Sigma\cup\=\tilde{\Sigma}$$. Every inner node (except root) has at least two children. Every edge is labeled with a string of $$\tilde{\Sigma}^+$$, labels of leaving edges at a single node start with different symbols and each leaf is indexed with $$1\dots m$$. The concatenation of all edge labels on a path from root to a leaf with index $$i$$ represents the suffix $$S[i\dots m]$$. Definition 1: For each node $$n$$: The path from root to $$n$$ is called $$p$$ • The union of the labels at all edges on $$p$$ is $$\alpha\in\tilde{\Sigma}^*$$ • $$\alpha$$ is label of path $$p$$ ($$\alpha=\text{label}(p)$$) • $$\alpha$$ is path label to $$n$$ ($$\alpha=\text{pathlabel}(n)$$) • instead of $$n$$ we can call this node $$\overline \alpha$$ Definition 2: A pattern $$\alpha\in\Sigma^*$$ exists in suffix tree of $$S\in\Sigma^+$$ (further called $$ST(S)$$) if and only if there is a $$\beta\in\tilde{\Sigma}^*$$, so that $$ST(S)$$ contains a node $$n$$ with $$\alpha\beta=\text{pathlabel}(n)$$ ($$n=\overline{\alpha\beta}$$). Definition 3: A substring $$\alpha$$ ends in node $$\overline \alpha$$ or in an edge to $$\overline{\alpha\beta}$$ with $$\beta\in\tilde{\Sigma}^*$$. Definition 4: A edge to a leaf is called leaf-edge. The tree $$ST(ababc)$$ contains all suffixes of the word $$ababc$$ extended with $$\$$$. This tree is visualized in figure 1.

## Building suffix trees: Write only top down

The write-only, top-down (WOTD) algorithm constructs the suffix tree in a top-down fashion.

### Algorithm

Let $$\overline \alpha$$ be a node in $$ST(S)$$, then $$\alpha$$ denotes the concatenation of all edge labels on the path to $$\overline \alpha$$ ($$\alpha=\text{pathlabel}(\overline \alpha)$$). Each node $$\overline \alpha$$ in the suffix tree represents the set of all suffixes that have the prefix $$\alpha$$. So the set of pathlabels to leafs below $$\overline \alpha$$ can be written as $$R(\overline \alpha)=\{\beta|\beta \in \tilde{\Sigma}^* \wedge \alpha\beta \text{ is suffix of }S\}$$ (all suffixes of the set of suffixes that start with $$\alpha$$).

This set is splitted in equivalence classes for each symbol $$c$$ with $$G(\overline \alpha, c)=\{c\beta\|c\in \tilde{\Sigma}\wedge c\beta \in R(\overline \alpha)\}$$ is the $$c$$-group of $$R(\overline \alpha)$$.

Case 1: For groups $$G(\overline \alpha, c)$$ that contain only one suffix $$\beta$$ we create a leaf $$\overline{\alpha\beta}$$ with the index $$|S| - |\alpha\beta|$$ and connect it to $$\overline \alpha$$ with an edge containing label $$\beta$$. Case 2: In groups $$G(\overline \alpha, c)$$ with a size of at least two we compute their longest common prefix $$\gamma$$ that starts with $$c$$ and create a node $$\overline{\alpha\gamma}$$. The connecting edge between $$\overline \alpha$$ and $$\overline{\alpha\gamma}$$ gets the label $$\gamma$$ and we continue recursively with this algorithm in node $$\overline{\alpha\gamma}$$ with $$R(\overline{\alpha\gamma})=\{\beta|\beta \in \tilde{\Sigma}^* \wedge \alpha\gamma\beta \text{ is suffix of }S\}=\{\beta|\gamma\beta\in R(\overline \alpha)\}$$

## Applications

### Exact pattern matching

All paths from the root of the suffix tree are labeled with the prefixes of path labels. That is, they’re labeled with prefixes of suffixes of the string $$S$$. Or, in other words, they’re labeled with substrings of $$S$$. To search for a pattern $$\delta$$ in $$S$$, just go through $$ST(S)$$, following paths labeled by the characters of $$\delta$$. At any node $$\overline\alpha$$ with $$\alpha$$ is prefix of $$\delta$$ find the edge with label $$\beta$$ that starts with symbol $$\delta[|\alpha|+1]$$. If such an edge doesn’t exists, $$\delta$$ isn’t a substring of $$S$$. Otherwise try to match the pattern with $$\beta$$ to node $$\overline{\alpha\beta}$$. If $$\alpha\beta$$ is not a prefix of $$\delta$$ you’ll either get a mismatch denoting that $$\delta$$ isn’t a substring of $$S$$, or you ran out of caracters of $$\delta$$ and found it in the tree. If $$\alpha\beta$$ is a prefix of $$\delta$$ continue searching at node $$\overline{\alpha\beta}$$. If you were able to find $$\delta$$ in $$ST(S)$$, $$S$$ contains $$\delta$$ at any position denoted by the indexes of leafs below your point of discovery.

### Minimal unique substrings

$$\alpha$$ is a minimal unique substring if and only if $$S$$ contains $$\alpha$$ exactly once and any prefix $$\beta$$ of $$\alpha$$ can be found at least two times in $$S$$.

To find such a minimal unique substring walk through the tree to nodes $$\overline\alpha$$ with a leaf-edge $$f$$. A minimal unique substring is $$\alpha c$$ with $$c$$ is the first char of $$label(f)$$, because its prefix $$\alpha$$ isn’t unique ($$\overline\alpha$$ has at least two leaving edges) and every extended version $$\alpha c \beta$$ has a prefix that is also unique.

### Maximal repeats

A maximal pair is a tuple $$(p_1,p_2,l)$$, so that $$S[p_1\dots p_1 + l - 1] = S[p_2\dots p_2 + l - 1]$$, but $$S[p_1-1] \neq S[p_2-1]$$ and $$S[p_1+l] \neq S[p_2+l]$$. A maximal repeat is the string represented by such tuple. If $$\alpha$$ is a maximal repeat there is a node $$\overline\alpha$$ in $$ST(S)$$. To find the maximal repeats do a DFS on the tree. Label each leaf with the left character of the suffix that it represents. For each internal node:

• If at least one child is labeled with c, then label it with c
• Else if its children’s labels are diverse, label with c.
• Else then all children have same label, copy it to current node.

Path labels to left-diverse nodes are maximal repeats.

### Generalized suffix trees

An extension of suffix trees are generalized suffix trees. With it you can represent multiple words in one single tree. Of course you have to modify the tree, so that you know which leaf index corresponds to which word. Just a little bit more to store in the leafs ;) A generalized suffix tree is printed in figure 2 of page one.

### Further applications

There are a lot of other applications for a suffix tree structure. For example finding palindromes, search for regular expressions, faster computing of the Levenshtein distance, data compression and so on…

## Implementation

I’ve implemented a suffix tree in Java. The tree is constructed via WOTD and finds maximal repeats and minimal unique substrings. I also wanted pictures for this post, thus, I added a functionality that prints GraphViz code that represents the tree.

Download: Java: Code from GitHub (Please take a look at the man-page. Browse bugs and feature requests.)