Some computational results on the efficiency of an exterior point algorithm
|
|
- Φερενίκη Ζάνος
- 6 χρόνια πριν
- Προβολές:
Transcript
1 1 Some computational results on the efficiency of an exterior point algorithm El-Said adr 1, K. Paparrizos, aloukas Thanasis, G. Varkas University of Macedonia, Department of Applied Informatics, 156 Egnatia Str., Thessaloniki, {it02185, thanasis, paparriz Abstract The aim of this paper is to present some computational results for the exterior point simplex algorithm (EPSA). Our implementation was carried out under the C environment. This algorithm seems to be more efficient than the classical primal simplex algorithm (PSA), employing Dantzig s pivoting rule. Preliminary computational studies on randomly generated sparse linear programs support this belief. Also we use a modification of a recently developed procedure for updating inverse matrices and incorporate it with EPSA. This calculation requires Θ (m 2 ) operations.in our computational study we use the modification of the product form of the inverse which is faster than the classical product form of the inverse.. Keywords: Linear programming: Exterior point algorithm; Algorithm evaluation; Numerical experiments 1 Research supported by the Greek Scholarship Foundation, (I.K.Y).
2 2 Μερικά υπολογιστικά αποτελέσματα για την αποδοτικότητα ενός εξωτερικού αλγορίθμου σημείου El-Said adr, K. Paparrizos, aloukas Thanasis, G. Varkas University of Macedonia, Department of Applied Informatics, 156 Egnatia Str., Thessaloniki, {it02185, thanasis, paparriz Περίληψη Ο στόχος αυτού του άρθρου είναι να παρουσιαστούν μερικά υπολογιστικά αποτελέσματα για τον εξωτερικό μονοκατευθυντικό αλγόριθμο σημείου (EPSA). Η εφαρμογή μας πραγματοποιήθηκε στο πλαίσιο του περιβάλλοντος C. Αυτός ο αλγόριθμος φαίνεται να είναι αποδοτικότερος από τον κλασσικό πρωταρχικής σπουδαιότητας μονοκατευθυντικό αλγόριθμο (PSA), υιοθέτηση Dantzig s να περιστρέψει κανόνας. Οι προκαταρκτικές υπολογιστικές μελέτες για τα τυχαία παραγμένα αραιά γραμμικά προγράμματα υποστηρίζουν αυτήν την πεποίθηση. Επίσης χρησιμοποιούμε μια τροποποίηση μιας πρόσφατα αναπτυγμένης διαδικασίας για τις αντίστροφες μήτρες και την ενσωματώνουμε με EPSA. Αυτός ο υπολογισμός απαιτεί Θ (m 2 ) διαδικασίες. Ι ν η υπολογιστική μελέτη μας χρησιμοποιούμε την τροποποίηση της μορφής προϊόντων του αντιστρόφου που είναι γρηγορότερο από την κλασσική μορφή προϊόντων του αντιστρόφου.. Λέξεις κλειδιά: Γραμμικός προγραμματισμός: Εξωτερικός αλγόριθμος σημείου; Αξιολόγηση αλγορίθμου; Αριθμητικά πειράματα
3 3 1. Introduction The simplex method has been studied extensively since its invention in 1947 by G.. Dantzig and still remains one of the most efficient methods for solving a great majority of practical problems. Although a number of variants of the simplex method have been developed, none of them has polynomial time complexity [16]. That is, the simplex method may require computational effort which grows exponentially with the size of the given problem. The first polynomial algorithm was invented by Khachian in 1979 and is called the ellipsoid algorithm.the practical performance of this algorithm is poor for practical problems compared with the simplex method. In 1984, Karmarkar presented a new polynomial algorithm for the linear problems (LP) which approaches the optimal solution from the interior of the feasible region [7]. Since this Karmarkar s seminal paper was published, many papers have been written about interior point methods [14, 17]. As their theoretical properties show, interior point methods work well for real world problems in real life and outperform the simplex method for very large-scale problems [8]. EPSA was originally developed by Paparrizos [9] for assignment problem. Latter, Paparrizos [10] generalized his exterior point method to the general linear problem by developing a dual, in nature, algorithm. Independently, Anstreicher and Terlaky [1], developed a similar primal algorithm solving a sequence of linear sub-problems. Another relevant result was developed by Chen et al. [4]. A common feature of almost all-simplex type algorithms is that they can be interpreted as a procedure following simplex type paths that lead to the optimal solution. This algorithm differs radically from the Primal Simplex Algorithm (PSA) because its basic solutions are not feasible. It has been pointed out by Paparrizos et al. [12] that the geometry of EPSA reveals that this algorithm is faster than the well-known simplex method, a fact that was verified by preliminary computational results comparing early dual versions of EPSA on specially structured linear problems (see[6]). Recently, Paparrizos et al.[13] presented computational results relating that EPSA is more efficient than the classical primal simplex algorithm PSA. In this paper, our aim is to study the standard linear programming form. In section 2, we recall some well-known facts about the simplex algorithm in revised form. As it is the first algorithm that solves linear programs it is very well studied in the literature. Also, we present the EPSA and a modification of a recently developed procedure for updating inverse matrices and incorporate it with exterior point algorithm. In Section 3 we give the computational results that demonstrate the superiority of our algorithm over simplex algorithm on a class
4 4 of randomly generated problem. Finally, in Section 4 we give our conclusions and discuss possible extensions of the algorithm. 2. Algorithm description We concerned with the following linear programming program minimize T c x subject to Ax = b (P.1) x 0, m n n m where A A, c, x, b and T denotes transposition. We find it convenient to describe the algorithm using the revised form of the simplex algorithm. Let us first assume that A is full rank, rank(a)=m, 1 m n. We will follow a time-honored traditional abuse of notation and write instead of A and N instead of AN. Whether and N stands for a subset of indices or a matrix should be clear for the context. With the matrix A partitioned as A= ( N ) and with a corresponding partitioning and ordering of x and c x x = x N c, c = (P.1) is written as c N min T cx + c T N x N subject to x + Nx N = b x, x 0 N Here is an m m non-singular submatrix of A called basic matrix. The columns of A which belong to are called basic and the remaining ones are 1 called non-basic. Given a basic the associated solution x = b, x N = 0 is called a basic solution. A solution x=( x, x N ) is feasible if x 0. Otherwise it is called infeasible. It is well known that the solution of the dual T problem, which corresponds to the basis, is given by s = c A w where T T 1 w = ( c ) are the simplex multipliers and s are the dual slack variables. The associated basis is called dual feasible if s 0. It is well known that s = 0.
5 5 In every iteration PSA interchanges a column of the current with a column of the current non-basic N, constructing in this way a new basis. Geometrically, this means that PSA moves along the edges of the polyhedron P = { x Ax b, x 0}. Such a path is known as a simplex path. Οn the other hand EPSA generates two paths toward the optimal solution. One path is infeasible (exterior) and the other is feasible. So EPSA does not need to proceed by following one such edge after another along the polyhedron. Therefore, avoiding the feasible region we can follow shorter paths to the optimal solution. Note that the total work of one iteration of simplex type algorithms is dominated 1 by the determination of the inverse matrix. This inverse however, does not 1 ( 1) have to be computed from scratch in each iteration. The next inverse ( ) r + 1 ( ) can be computed from the current inverse ( ) r with a simple pivoting operation. Namely we have ( ) = E ( ) 1 ( r+ 1) 1 1 ( r) 1 E E 1 Where is the inverse of an eta-matrix. We use relation (2) to compute. 1 1 ( ) T E = I aq eq e q (2) a pq a pq In the above relation denotes the pivot element, column q is called the pivot column and row p is called the pivot row. A formal description of the primal EPSA in the revised form is given below. The above technique is called product form of the inverse. (1) 2.1. EPSA algorithm Step 0 (Initialization). Start with a feasible basic partition (,N). Compute the vectors and matrices 1, x, w, s N. Find the sets P = { j N : s j <0} and Q = { j N s j 0} choose an arbitrary vector λ = ( λ1, λ2,..., λ p ) > 0, compute s 0 using the relation s 0 = λ j s j and the vector d = λ jhj with j P 1 hj = A. j j P
6 = 6 Step 1 (Test of termination) (i) (Test of optimality) If P =, STOP. Problem (P.1) is optimal. (ii) (Choice of leaving variable). If d 0, STOP. If s0 = 0, problem (P.1) is optimal. Otherwise, choose the leaving variable x [ r] = x k using the relation x [ r] α = = d [ r] min x d [ i ] [ i ] If α =+ problem (P.1) is unbounded. Put : d [ i ] < 0 y = x + α d Step 2 (Choice of entering variable). 1 1 Compute the vectors H rp = ( ) r. A. P and H rq = ( ) r. A. Q. Also find the ratios θ1 and θ 2 using the relations θ θ s s P j 1 = =min : rj 0 and hrp hrj s s h > j P Q j 2 = =min : rj 0 and hrq hrj h < j Q and determine indexes t 1 and t 2 such that P(t 1 ) = p and Q(t 2 ) = q. If θ1 θ 2, set l = p. Otherwise, set l = q.the non-basic variable x l enter in the basis. Step 3 (Pivoting). Set [r] = l. If θ 1 θ 2 set P = P \{ l} and Q = Q { k}. Otherwise, set Qt [] = k. Using the new partition (, N), where N =(P, Q), update the vectors and matrices -1, x, w, sn, d y - x. Go to step 1.
7 7 2.2 A Modification of Product Form of the Inverse To justify the next algorithm and analysis we need the following notations: x y : outer product of the vectors x, y R n l : index of the entering variable k : index of the leaving variable A. l : the lth column of A -1 r. : the rth row of basis inverse -1 r. : the matrix -1 with the rth row put to zero This updating scheme presented by [3]. The key idea is the following. The current basis inverse 1-1 can be computed from the previous inverse with a simple outer product of two vectors and one matrix addition. Namely we have 1 1 = + v 1 (3) r. r. A formal description of this method follows. -1 Compute the pivot column h l = A. l Compute the vector h1l 1 h v = h h h Compute the outer product v Set the rth row of ml rl rl rl -1 r. -1 equal to zero. Save the result in Compute the new basis inverse using relation (3). T -1 r. The outer product requires m 2 multiplications and the addition of two matrices requires m 2 additions. The total cost of the above method is 2m 2 operations (multiplications and additions). Hence, the complexity is Θ(m 2 ). The computational study indicates that the modification of the product form of the inverse is faster than the product form of the inverse (adr et al. 2005) so that we use the modification of the product form of the inverse in our implementation.
8 8 3. Computational results The algorithm described in Section 2 have been experimentally implemented. In this section we describe our numerical experiments and present computational results, which demonstrate EPSA s efficiency on random sparse linear programs. In this computational analysis we perform a comparison between PSA and EPSA. Our numerical experiments were performed on a PC with MHz Pentium 4 processor, RAM 512 Mb and Windows XP operating system. Our implementation was done under the C environment. Table 1 Computational results for sparse linear programs n n (CPU in seconds) n n nnz PSA EPSA niter CPU niter CPU Density 5% Density 10% We run total of 280 random sparse problems of density 5% and 10%. The problems we tested were small and medium in size.
9 9 Table 2 Computational results for sparse linear programs n 2n (CPU in seconds) n 2n nnz PSA EPSA niter CPU niter CPU Density 5% Density 10% Table 3 Computational results for sparse linear programs 2n n (CPU in seconds) 2n n nnz PSA EPSA niter CPU niter CPU Density 5% Density 10% The sparse linear optimization problems that have been solved are of the form minimize T c x subject to Ax = b x 0,
10 10 m n n m where A, c, x, b and T denotes transposition and an intial feasible solution exists. In our computational study we use two different density cases of the problems: 5% and 10%. For each density case there are three different dimension cases : n n, n 2n and 2n n. The first case n n, include eight different classes of problems corresponding to the values n = 300, 350, 400, 450, 500, 600, 700,800; each of these classes contains ten random sparse linear programs. Those of the other two cases n 2n and 2n n, include three different classes of problems corresponding to the values n = 300, 350, 400. The ranges of values that were used for randomly generated linear programs for all densities are c [1 500], A [ ], r = 7 and center=35. For each size and density of the problem we bring together some statistics on the LPs used in our computational study and information on the performance of the two algorithms, PSA and EPSA. Columns of Table 1-3 contain problems size, number of non-zero elements of the constraint matrix ( excluding cost and right-hand size vector) nnz, the average number of iterations, niter and the mean CPU time, CPU. In order to show more clearly the superiority of EPSA over PSA we provide now some tables showing for each case of density relative with the above tables. Table 4 Ratios for sparse linear programs n n n n niter PSA/EPSA CPU PSA/EPSA Density 5% Mean value
11 11 Density 10% Mean value Table 5 Ratios for sparse linear programs 2n n n n niter PSA/EPSA CPU PSA/EPSA Density 5% Mean value Density 10% Mean value Table 6 Ratios for sparse linear programs n 2n n n niter PSA/EPSA CPU PSA/EPSA Density 5% Mean value Density 10% Mean value
12 12 In Tables 4-6 were present the ratios (iterations of PSA)/(iterations of EPSA) and (CPU time of PSA)/(CPU time of EPSA) for the corresponding densities and dimensions. At the end of each density case we give the mean values of the above ratios for each class of test problem. We now plot the ratios taken from Table 4 (Fig.1). The plot is for the sparse n n liner programs for all density cases. ratios for optimal of density 5% ratios optimal LPs off density10% ratios niter CPU ratios niter CPU problem size nxn problem size, nxn Fig.1 Ratios of PSA over EPSA for optimal sparse problems n n From the plot above we can see that as the problem dimension increases the superiority of EPSA over PSA increases. We can see for example, in case of sparse LPs of dimension n n that as the problem density decreases, the superiority of our algorithm over PSA increases. This conclusion is crucial because all the real life problems are extremely sparse. Particularly, in dimension of density 5%, EPSA is times faster than PSA in terms of iterations and solves the problems times faster than PSA in terms of CPU time. Although the computational effort required in each step of EPSA requires more time compared to n iteration step of PSA, the improvement of EPSA comes from the fact that it requires adequately less iterations than PSA. Moreover, as the problem size increase and the problem density decrease, EPSA gets relatively faster. 4. Conclusion In this paper we have resented a computational attractive approach to solving sparse linear programming problems. In the previous section we have addressed the most issue of an efficient implementation of our algorithm EPSA. The computational study of Section 3 indicates that as the problem size increases
13 13 and the density of problem decreases the frequency when EPSA outperforms the simplex algorithm increases. In our opinion this is a good practical performance that gives much promise to the approach. EPSA s algorithm experimental implementation in randomly generated LPs proved that the algorithm is fast; there is still room for its further improvement. These possible improvements will be the subject of our future work. References: [1]Anstreich, G.,Terlaky, T. A monotonic build-up simplex algorithm for linear programming,operation Research 42 (1994) [2]adr, E., Paparrizos, K., Samaras, N., Sifaleras, A. (2005).On the basis inverse of the exterior point simplex algorithm.17 ο Εθνικό Συνέδριο Ελληνικής Εταιρίας ΕπιχειρησιακώνΕρευνών. [3]enhamadou, M. (2002). On the simplex algorithm revised form, Advances in Engineering Software, 33, [4]Chen, H.,Pardalos, P. Saunders, M. The simplex algorithm with a new primal and dual pivot rule, Operations Research etters 16 (1994) [5]Dantzig, G.. Linear programming and extensions. Princeton, NJ: Princeton University Press; [6]Dosios K., Paparrizos K., Resolution of problem of degeneracy in a primal and dul simplex algorithm, Operation Research Letters 20 (1997) [7] Karmarkar, N.K. (1984).A new polynomial-time algorithm for linear programming. Combinatorica, 4(4), [8] Lustig, L.J., Marsten R.E. and Shanno, D.F. (1994). Interior point methods for linear programming: computational state of the art. ORSA Journal on Computing, 6(1), [9]Paparrizos, K. An infeasible exterior point simplex algorithm for assignment problems,mathematical Programming 51 (1991) [10]Paparrizos, K. A generalization of an exterior point simplex algorithm for linear programming problems, Technical Report,University of Macedonia, [11]Paparrizos, K. An exterior point simplex algorithm for general linear problems, nnals of Operation Research 47 (1993)
14 14 [12]Paparrizos, K., Samaras, N., Tsiplidis, K. Pivoting algorithm for (LP) generating two paths,in: M. P. Pardalos, A. C. Fluodas (Eds.), Encyyclopedia of Optimization, vol 4, Kluwer Academic Publishers, 2001, pp [13]Paparrizos, K., Samaras, Stephanides G., An efficient simplex type algorithm for sparse and dense linear programs, European Journal of Operational Research 148 (2003) [14] Roos, C., Terlaky, T. and J.-Ph. Vial (1997).Theory and Algorithms for Linear Optimization. John Wiley & Sons, New York, USA. [15]Strang, G. Introduction to applied mathematics. Wellesley, MA: Wellesley-Cambridge Press; [16] Terlaky, T.and Zhang, S. (1993). Pivot rules for linear programming: a survey on recent theoretical developments. Annals of Operations Research, 46, [17] Wright,S.J.(1997).Primal-Dual Interior-Point Methods.SIAM, Philadelphia.
Παράλληλος προγραμματισμός περιστροφικών αλγορίθμων εξωτερικών σημείων τύπου simplex ΠΛΟΣΚΑΣ ΝΙΚΟΛΑΟΣ
Παράλληλος προγραμματισμός περιστροφικών αλγορίθμων εξωτερικών σημείων τύπου simplex ΠΛΟΣΚΑΣ ΝΙΚΟΛΑΟΣ Διπλωματική Εργασία Μεταπτυχιακού Προγράμματος στην Εφαρμοσμένη Πληροφορική Κατεύθυνση: Συστήματα Υπολογιστών
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραΕΥΧΑΡΙΣΤΙΕΣ. Θεσσαλονίκη, Δεκέμβριος 2005. Κώστας Δόσιος
ΕΥΧΑΡΙΣΤΙΕΣ Μου δίνεται η ευκαιρία με την περάτωση της παρούσης διδακτορικής διατριβής να σημειώσω ότι, είναι ιδιαίτερα δύσκολο και κοπιαστικό να ολοκληρώσεις το έργο που ξεκινάς κάποια στιγμή έχοντας
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραΕΝΑΣ ΝΕΟΣ ΤΡΟΠΟΣ ΔΗΜΙΟΥΡΓΙΑΣ ΤΥΧΑΙΩΝ ΒΕΛΤΙΣΤΩΝ ΓΡΑΜΜΙΚΩΝ ΠΡΟΒΛΗΜΑΤΩΝ ΚΑΙ ΜΙΑ ΣΥΓΚΡΙΤΙΚΗ ΥΠΟΛΟΓΙΣΤΙΚΗ ΜΕΛΕΤΗ
ΕΝΑΣ ΝΕΟΣ ΤΡΟΠΟΣ ΔΗΜΙΟΥΡΓΙΑΣ ΤΥΧΑΙΩΝ ΒΕΛΤΙΣΤΩΝ ΓΡΑΜΜΙΚΩΝ ΠΡΟΒΛΗΜΑΤΩΝ ΚΑΙ ΜΙΑ ΣΥΓΚΡΙΤΙΚΗ ΥΠΟΛΟΓΙΣΤΙΚΗ ΜΕΛΕΤΗ Παπαρρίζος Κωνσταντίνος, Σαμαράς Νικόλαος, Στεφανίδης Γεώργιος Τμ. Εφαρμοσμένης Πληροφορικής
Διαβάστε περισσότεραInstruction Execution Times
1 C Execution Times InThisAppendix... Introduction DL330 Execution Times DL330P Execution Times DL340 Execution Times C-2 Execution Times Introduction Data Registers This appendix contains several tables
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραTMA4115 Matematikk 3
TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραPhysical DB Design. B-Trees Index files can become quite large for large main files Indices on index files are possible.
B-Trees Index files can become quite large for large main files Indices on index files are possible 3 rd -level index 2 nd -level index 1 st -level index Main file 1 The 1 st -level index consists of pairs
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ
ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΨΥΧΟΛΟΓΙΚΕΣ ΕΠΙΠΤΩΣΕΙΣ ΣΕ ΓΥΝΑΙΚΕΣ ΜΕΤΑ ΑΠΟ ΜΑΣΤΕΚΤΟΜΗ ΓΕΩΡΓΙΑ ΤΡΙΣΟΚΚΑ Λευκωσία 2012 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ
Διαβάστε περισσότεραOn a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραNew bounds for spherical two-distance sets and equiangular lines
New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a
Διαβάστε περισσότερα5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΒΑΛΕΝΤΙΝΑ ΠΑΠΑΔΟΠΟΥΛΟΥ Α.Μ.: 09/061. Υπεύθυνος Καθηγητής: Σάββας Μακρίδης
Α.Τ.Ε.Ι. ΙΟΝΙΩΝ ΝΗΣΩΝ ΠΑΡΑΡΤΗΜΑ ΑΡΓΟΣΤΟΛΙΟΥ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ «Η διαμόρφωση επικοινωνιακής στρατηγικής (και των τακτικών ενεργειών) για την ενδυνάμωση της εταιρικής
Διαβάστε περισσότεραCHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3
Διαβάστε περισσότεραΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ "ΠΟΛΥΚΡΙΤΗΡΙΑ ΣΥΣΤΗΜΑΤΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ. Η ΠΕΡΙΠΤΩΣΗ ΤΗΣ ΕΠΙΛΟΓΗΣ ΑΣΦΑΛΙΣΤΗΡΙΟΥ ΣΥΜΒΟΛΑΙΟΥ ΥΓΕΙΑΣ "
ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΚΑΛΑΜΑΤΑΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΜΟΝΑΔΩΝ ΥΓΕΙΑΣ ΚΑΙ ΠΡΟΝΟΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ "ΠΟΛΥΚΡΙΤΗΡΙΑ ΣΥΣΤΗΜΑΤΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ. Η ΠΕΡΙΠΤΩΣΗ ΤΗΣ ΕΠΙΛΟΓΗΣ ΑΣΦΑΛΙΣΤΗΡΙΟΥ ΣΥΜΒΟΛΑΙΟΥ
Διαβάστε περισσότεραΑΠΟΔΟΤΙΚΗ ΑΠΟΤΙΜΗΣΗ ΕΡΩΤΗΣΕΩΝ OLAP Η ΜΕΤΑΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΕΞΕΙΔΙΚΕΥΣΗΣ. Υποβάλλεται στην
ΑΠΟΔΟΤΙΚΗ ΑΠΟΤΙΜΗΣΗ ΕΡΩΤΗΣΕΩΝ OLAP Η ΜΕΤΑΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΕΞΕΙΔΙΚΕΥΣΗΣ Υποβάλλεται στην ορισθείσα από την Γενική Συνέλευση Ειδικής Σύνθεσης του Τμήματος Πληροφορικής Εξεταστική Επιτροπή από την Χαρά Παπαγεωργίου
Διαβάστε περισσότεραΓραµµικός Προγραµµατισµός (ΓΠ)
Γραµµικός Προγραµµατισµός (ΓΠ) Περίληψη Επίλυση δυσδιάστατων προβληµάτων Η µέθοδος simplex Τυπική µορφή Ακέραιος Προγραµµατισµός Προγραµµατισµός Παραγωγής Προϊόν Προϊόν 2 Παραγωγική Δυνατότητα Μηχ. 4 Μηχ.
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραA Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics
A Bonus-Malus System as a Markov Set-Chain Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics Contents 1. Markov set-chain 2. Model of bonus-malus system 3. Example 4. Conclusions
Διαβάστε περισσότεραNotes on the Open Economy
Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4.
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραΠΑΡΑΜΕΤΡΟΙ ΕΠΗΡΕΑΣΜΟΥ ΤΗΣ ΑΝΑΓΝΩΣΗΣ- ΑΠΟΚΩΔΙΚΟΠΟΙΗΣΗΣ ΤΗΣ BRAILLE ΑΠΟ ΑΤΟΜΑ ΜΕ ΤΥΦΛΩΣΗ
ΠΑΝΕΠΙΣΤΗΜΙΟ ΜΑΚΕΔΟΝΙΑΣ ΟΙΚΟΝΟΜΙΚΩΝ ΚΑΙ ΚΟΙΝΩΝΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΕΚΠΑΙΔΕΥΤΙΚΗΣ ΚΑΙ ΚΟΙΝΩΝΙΚΗΣ ΠΟΛΙΤΙΚΗΣ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΑΡΑΜΕΤΡΟΙ ΕΠΗΡΕΑΣΜΟΥ ΤΗΣ ΑΝΑΓΝΩΣΗΣ- ΑΠΟΚΩΔΙΚΟΠΟΙΗΣΗΣ ΤΗΣ BRAILLE
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότερα9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότεραStrain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Διαβάστε περισσότεραΜηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραw o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότεραΚΑΘΟΡΙΣΜΟΣ ΠΑΡΑΓΟΝΤΩΝ ΠΟΥ ΕΠΗΡΕΑΖΟΥΝ ΤΗΝ ΠΑΡΑΓΟΜΕΝΗ ΙΣΧΥ ΣΕ Φ/Β ΠΑΡΚΟ 80KWp
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΤΟΜΕΑΣ ΣΥΣΤΗΜΑΤΩΝ ΜΕΤΑΔΟΣΗΣ ΠΛΗΡΟΦΟΡΙΑΣ ΚΑΙ ΤΕΧΝΟΛΟΓΙΑΣ ΥΛΙΚΩΝ ΚΑΘΟΡΙΣΜΟΣ ΠΑΡΑΓΟΝΤΩΝ ΠΟΥ ΕΠΗΡΕΑΖΟΥΝ ΤΗΝ ΠΑΡΑΓΟΜΕΝΗ ΙΣΧΥ
Διαβάστε περισσότεραProblem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότεραJordan Form of a Square Matrix
Jordan Form of a Square Matrix Josh Engwer Texas Tech University josh.engwer@ttu.edu June 3 KEY CONCEPTS & DEFINITIONS: R Set of all real numbers C Set of all complex numbers = {a + bi : a b R and i =
Διαβάστε περισσότεραCommutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότερα( ) 2 and compare to M.
Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8
Διαβάστε περισσότεραHomomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
Διαβάστε περισσότεραAssalamu `alaikum wr. wb.
LUMP SUM Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. LUMP SUM Lump sum lump sum lump sum. lump sum fixed price lump sum lump
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραAbstract Storage Devices
Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD
Διαβάστε περισσότεραΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΓΕΩΤΕΧΝΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΚΑΙ ΔΙΑΧΕΙΡΙΣΗΣ ΠΕΡΙΒΑΛΛΟΝΤΟΣ. Πτυχιακή εργασία
ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΓΕΩΤΕΧΝΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΚΑΙ ΔΙΑΧΕΙΡΙΣΗΣ ΠΕΡΙΒΑΛΛΟΝΤΟΣ Πτυχιακή εργασία ΑΝΑΛΥΣΗ ΚΟΣΤΟΥΣ-ΟΦΕΛΟΥΣ ΓΙΑ ΤΗ ΔΙΕΙΣΔΥΣΗ ΤΩΝ ΑΝΑΝΕΩΣΙΜΩΝ ΠΗΓΩΝ ΕΝΕΡΓΕΙΑΣ ΣΤΗΝ ΚΥΠΡΟ ΜΕΧΡΙ ΤΟ 2030
Διαβάστε περισσότεραAΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΠΟΛΙΤΙΚΩΝ ΜΗΧΑΝΙΚΩΝ
AΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΠΟΛΙΤΙΚΩΝ ΜΗΧΑΝΙΚΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΕΙΔΙΚΕΥΣΗΣ ΠΡΟΣΤΑΣΙΑ ΠΕΡΙΒΑΛΛΟΝΤΟΣ ΚΑΙ ΒΙΩΣΙΜΗ ΑΝΑΠΤΥΞΗ ΔΙΕΡΕΥΝΗΣΗ ΤΩΝ ΠΙΕΣΕΩΝ ΣΤΟ ΠΕΡΙΒΑΛΛΟΝ
Διαβάστε περισσότεραA Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότεραQuick algorithm f or computing core attribute
24 5 Vol. 24 No. 5 Cont rol an d Decision 2009 5 May 2009 : 100120920 (2009) 0520738205 1a, 2, 1b (1. a., b., 239012 ; 2., 230039) :,,.,.,. : ; ; ; : TP181 : A Quick algorithm f or computing core attribute
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραThe challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
Διαβάστε περισσότεραΣυστήματα Διαχείρισης Βάσεων Δεδομένων
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Συστήματα Διαχείρισης Βάσεων Δεδομένων Φροντιστήριο 9: Transactions - part 1 Δημήτρης Πλεξουσάκης Τμήμα Επιστήμης Υπολογιστών Tutorial on Undo, Redo and Undo/Redo
Διαβάστε περισσότεραOverview. Transition Semantics. Configurations and the transition relation. Executions and computation
Overview Transition Semantics Configurations and the transition relation Executions and computation Inference rules for small-step structural operational semantics for the simple imperative language Transition
Διαβάστε περισσότεραΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΜΗΧΑΝΙΚΩΝ Η/Υ & ΠΛΗΡΟΦΟΡΙΚΗΣ. του Γεράσιμου Τουλιάτου ΑΜ: 697
ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΜΗΧΑΝΙΚΩΝ Η/Υ & ΠΛΗΡΟΦΟΡΙΚΗΣ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΣΤΑ ΠΛΑΙΣΙΑ ΤΟΥ ΜΕΤΑΠΤΥΧΙΑΚΟΥ ΔΙΠΛΩΜΑΤΟΣ ΕΙΔΙΚΕΥΣΗΣ ΕΠΙΣΤΗΜΗ ΚΑΙ ΤΕΧΝΟΛΟΓΙΑ ΤΩΝ ΥΠΟΛΟΓΙΣΤΩΝ του Γεράσιμου Τουλιάτου
Διαβάστε περισσότεραΦΩΤΟΓΡΑΜΜΕΤΡΙΚΕΣ ΚΑΙ ΤΗΛΕΠΙΣΚΟΠΙΚΕΣ ΜΕΘΟΔΟΙ ΣΤΗ ΜΕΛΕΤΗ ΘΕΜΑΤΩΝ ΔΑΣΙΚΟΥ ΠΕΡΙΒΑΛΛΟΝΤΟΣ
AΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΠΟΛΙΤΙΚΩΝ ΜΗΧΑΝΙΚΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΕΙΔΙΚΕΥΣΗΣ ΠΡΟΣΤΑΣΙΑ ΠΕΡΙΒΑΛΛΟΝΤΟΣ ΚΑΙ ΒΙΩΣΙΜΗ ΑΝΑΠΤΥΞΗ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΦΩΤΟΓΡΑΜΜΕΤΡΙΚΕΣ
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραPotential Dividers. 46 minutes. 46 marks. Page 1 of 11
Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and
Διαβάστε περισσότεραdepartment listing department name αχχουντσ ϕανε βαλικτ δδσϕηασδδη σδηφγ ασκϕηλκ τεχηνιχαλ αλαν ϕουν διξ τεχηνιχαλ ϕοην µαριανι
She selects the option. Jenny starts with the al listing. This has employees listed within She drills down through the employee. The inferred ER sttricture relates this to the redcords in the databasee
Διαβάστε περισσότεραCorrection Table for an Alcoholometer Calibrated at 20 o C
An alcoholometer is a device that measures the concentration of ethanol in a water-ethanol mixture (often in units of %abv percent alcohol by volume). The depth to which an alcoholometer sinks in a water-ethanol
Διαβάστε περισσότεραSpace-Time Symmetries
Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a
Διαβάστε περισσότερα