The CP T Theorem

INTRODUCTION TO PROBABILITY - CHAPTER 9 PPS

Figure 9.10: Distribution of heights of adult women. tend to have tall offspring. Thus in this case, there seem to be two large effects, namely the parents. Galton was certainly aware of the fact that non-genetic factors played a role in determining the height of an individual. Nevertheless, unless these non-genetic factors overwhelm the genetic ones, thereby refuting the hypothesis that heredity is important in determining height, it did not seem possible for sets of parents of given heights to have offspring whose heights were normally distributed. One can express the above problem symbolically as follows. Suppose that we choose two specific positive real numbers x and y , and then find all pairs of parents one of whom is x units tall and the other of whom is y units tall. We then look at all of the offspring of these pairs of parents. One can postulate the existence of a function f ( x, y ) which denotes the genetic effect of the parents’ heights on the heights of the offspring. One can then let W denote the effects of the non-genetic factors on the heights of the offspring. Then, for a given set of heights { x, y } , the random variable which represents the heights of the offspring is given by

DCSTRADEOFF DOCX

 Channel capacity of AWGV channel Shannon-Hartley capacity theorem: ] [bits/s 1 log2  TRANG 8 SHANNON LIMIT …  THE SHANNON THEOREM PUTS A LIMIT ON THE TRANSMISSION DATA RATE, NOT ON T[r]

TỪ ĐIỂN TOÁN HỌC ANH – VIỆT MẪU CÂU TOÁN HỌC ANH – VIỆT TỔNG HỢP NHỮNG THUẬT NGỮ TOÁN HỌC TỪ ĐIỂN TOÁN HỌC ANH – VIỆT DICTIONARY OF MATHEMATICS TERMS DICTIONARY OF MATHEMATICS DIC

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ELECTROMAGNETIC FIELD THEORY A PROBLEM SOLVING APPROACH PART 7 DOCX

circular area that is perpendicular to Vx A. The integral is then the same as part (a) as V X A is independent of z. 1-5-4 Some Useful Vector Identities The curl, divergence, and gradient operations have some simple but useful properties that are used throughout the text.

NETWORKING THEORY AND FUNDAMENTALS LECTURES 9 & 10 PPS

Proof: Identical to the PASTA theorem due to: Poisson arrivals Lack of anticipation: future arrivals independent of current state N(t) Theorem 3: For an M/G/1 queue at steady-state, the system appears statistically identical to an arriving and a departing customer. Both an arriving and a departing customer, at steady-state, see a system that is statistically identical to the one seen by an observer looking at the system at an arbitrary time.

BÁO CÁO HÓA HỌC RESEARCH ARTICLE LAGRANGIAN STABILITY OF A CLASS OF SECOND ORDER PERIODIC SYSTEMS DOCX

x F x, t x ω p Φ p x α | x | l x e x, t 0 . 1.9 where Φ p s | s | p −2 s p > 1 , − 1 < l < p − 2, and α, ω > 0 are constants. We want to generalize the result in 6 to a class of p -Laplacian-type di ﬀ erential equations of the form 1.9 . The main idea is similar to that in 6 . We will assume that the functions F and e have some parities such that the di ﬀ erential system 1.9 still has a reversible structure. After some transformations, we change the systems 1.9 to a form of small perturbation of integrable reversible system. Then a KAM Theorem for reversible mapping can be applied to the Poincar´e mapping of this nearly integrable reversible system and some desired result can be obtained.

BÁO CÁO TOÁN HỌC MOUNTAIN PASS THEOREM AND NONUNIFORMLY ELLIPTIC EQUATIONS DOC

exists u o ∈ Y such that J ( u 0 ) = β > 0. On the other hand, J (0) = 0. Hence, u 0 is a nontrivial critical point of J . Furthermore, since C c ∞ (Ω) ⊂ Y , the critical point u 0 of J is a nontrivial generalized solution to the problem ( P ). (ii) Let us assume that A and F are even with respect to the second variable. Then J is even. According to Lemmata 3.2 and 3.6, Theorem 1.3 can be applied to the function I ≡ J with E ≡ Y , V ≡ { 0 } . Hence, J possesses an unbounded sequence of critical values. Therefore, J possesses inﬁnitely many critical points in Y .

NETWORKING THEORY AND FUNDAMENTALS LECTURE 4 & 5 DOC

a n = d n , n= 0,1,… In steady-state, system appears stochastically identical to an arriving and departing customer Poisson arrivals + LAA: an arriving and a departing customer see a system that is stochastically to the one seen by an observer looking at an arbitrary time

Inverse version of the kth maximization combinatorial optimization problem

Theorem 4.2 The inverse

REAL ANALYSIS WITH ECONOMIC APPLICATIONS - HINTS REFERENCES DOC

Exercise 24. Q is countable and R\Q is not. So, by the Intermediate Value Theorem ...? Exercise 25. Punch a hole in R n . Now do the same in R . Do you get similar spaces? Exercise 27. You can easily prove this by using Theorem C.2 and imitating the way we proved Proposition A.11. Let me suggest a more direct argument here. Take any ε > 0 . Since f is continuous, for any x ∈ X there exists a δ x > 0 such that d Y ( f ( x ) , f ( y )) < ε 2 for all y ∈ X with d X ( x, y ) < δ x . Now use the compactness of X to fi nd fi nitely many x 1 , ..., x m ∈ X with X = V

BÁO CÁO TOÁN HỌC: "ON THE THEORY OF PFAFFIAN ORIENTATIONS. II. T -JOINS, K-CUTS, AND DUALITY OF ENUMERATION" POTX

Received February 18, 1998; Accepted May 22, 1998. Abstract This is a continuation of our paper “A Theory of Pfaffian Orientations I: Perfect Matchings and Permanents”. We present a new combinatorial way to compute the generating functions of T -joins and k-cuts of graphs. As a consequence, we show that the computational problem to find the maximum weight of an edge-cut is polynomially solvable for the instances (G, w) where G is a graph embedded on an arbitrary fixed orientable surface and the weight function w has only a bounded number of different values. We also survey the related results concerning a duality of the Tutte polynomial, and present an application for the weight enumerator of a binary code. In a continuation of this paper which is in preparation we present an application to the Ising problem of three-dimensional crystal structures.

INDEPENDENT AND STATIONARY SEQUENCES OF RANDOM VARIABLES CHAPTER 8 PPS

The first general result in the theory of large deviations was the integral theorem of Cramer [19], published in 1938, which has considerable computational and analytical usefulness . It was refined and generalised by Petrov [133] in 1954. In this chapter we discuss the work of Petrov, keeping for simplicity to the case of identically distributed variables X;. It should be remarked that the most natural method for proving integral theorems under Cramer's condition is the method of steepest descents, whose use for local theorems was described in the last chapter. In the case in which the distributions of the X; are different such an approach

BLAST INTO MATH! - EBOOKS AND TEXTBOOKS FROM BOOKBOON

A proposition, like a theorem, is a true mathematical statement, but it is usually easier to prove than a theorem, so we do not endow it with the hefty and distinguished title “theorem.”[r]

BÁO CÁO TOÁN HỌC: "UNIVERSALLY IMAGE PARTITION REGULARITY" POTX

2 Infinite Matrices The definition of universally image partition regularity has a natural generalization for the matrices of order ω × ω . We mention here that when we talk of an infinite matrix we shall assume that each row of it contains only finitely many nonzero elements. In the previous section we have seen that if a matrix with entries from ω is image partition regular over N then it is universally image partition regular. In this section we see that there are a lots of variety in the infinite case. First we observe that the finite sums matrix

BÁO CÁO TOÁN HỌC: "HYBRID PROOFS OF THE Q-BINOMIAL THEOREM AND OTHER IDENTITIES" PPSX

In the other camp there are a variety of combinatorial or bijective proofs. Rather than attempt any classification of the various bijective proofs, we refer the reader to Pak’s excellent survey [21] of bijective methods, with its extensive bibliography. In the present paper we use a “hybrid” method to prove a number of basic hyperge- ometric identities. The proofs are “hybrid” in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version.

BÁO CÁO HÓA HỌC SEVERAL FIXED POINT THEOREMS CONCERNING Τ DISTANCE POT

Remark 3.8. z ∈ Tz does not necessarily imply p ( z , z ) = 0; see Example 3.9 . Proof. By the assumption, there exists r ∈ [0, 1) such that Q ( Tx , T y ) ≤ r p ( x , y ) for all x , y ∈ X . Put r = (1 + r ) / 2 ∈ [0, 1) and fix x , y ∈ X and u ∈ Tx . Then, in the case of p ( x , y ) > 0, there is v ∈ T y satisfying p ( u , v ) ≤ r p ( x , y ). In the case of p ( x , y ) = 0, we have Q ( Tx , T y ) = 0. Then there exists a sequence { v n } in T y satisfying lim n p ( u , v n ) = 0. By Lemma 2.5 , { v n } is p -Cauchy, and hence { v n } is Cauchy. Since X is complete and T y

EBOOK A STUDENT’S GUIDE TO FOURIER TRANSFORMS (3RD EDITION) PART 2

9 Discrete and digital Fourier transforms 9.1 History Fourier transformation is formally an analytic process which uses integral cal- culus. In experimental physics and engineering, however, the integrand may be a set of experimental data, and the integration is necessarily done artifi- cially. Since a separate integration is needed to give each point of the trans- formed function, the process would become exceedingly tedious if it were to be attempted manually, and many ingenious devices have been invented for performing Fourier transforms mechanically, electrically, acoustically and opti- cally. These are all now part of history since the arrival of the digital computer and more particularly since the discovery – or invention – of the ‘fast Fourier transform’ algorithm or FFT as it is generally called. Using this algorithm, the data are put (‘read’) into a file (or ‘array’, depending on the computer jargon in use), the transform is carried out, and the array then contains the points of the transformed function. It can be achieved by a software program, or by a purpose-built integrated circuit. It can be done very quickly so that vibration- sensitive instruments with Fourier transformers attached can be used for tuning pianos and motor engines, for aircraft and submarine detection and so on. It must not be forgotten that the ear is Nature’s own Fourier transformer, 1 and, as used by an expert piano-tuner, for example, is probably the equal of any electronic simulator in the 20–20 000-Hz range. The diffraction grating, too, is a passive Fourier transformer device, provided that it is used as a spectrograph taking full advantage of the simultaneity of outputs.

TO 1 AS N

3 THE EXPECTED NUMBER OF T-STABLE SETS OF ORDER K -PROOF OF THEOREM 2 In this section, we give an asymptotic expression for the expected number of t-stable subsets of Vn of order k in Gn[r]

BÁO CÁO TOÁN HỌC: "THE T-STABILITY NUMBER OF A RANDOM GRAPH" PPTX

Abstract Given a graph G = ( V, E ), a vertex subset S ⊆ V is called t -stable (or t - dependent) if the subgraph G [ S ] induced on S has maximum degree at most t . The t -stability number α t ( G ) of G is the maximum order of a t -stable set in G . The theme of this paper is the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p . For any fixed 0 < p < 1 and fixed non-negative integer t , we show that, with probability tending to 1 as n → ∞ , the t -stability number takes on at most two values which we identify as functions of t , p and n . The main tool we use is an asymptotic expression for the expected number of t -stable sets of order k . We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most t .

BÁO CÁO TOÁN HỌC: "RELAXED GRACEFUL LABELLINGS OF TREES" PDF

5 Closing remarks 5.1 Better bounds Even if the graceful tree conjecture remains unattainable, better bounds for range-relaxed and vertex-relaxed graceful labellings of trees should be within reach. The techniques employed here, however, may not take us much further, since our results depend in part upon the tree’s diameter for RRG labellings and the difference in the size of its bipartition sets for VRG labellings; both may be arbitrarily small in relation to the size of the tree. Hence different approaches will probably need to be taken in order to obtain bounds that are, for example, comparable to those of Rosa and ˇ Sir´ aˇ n for edge-relaxed graceful labellings.

PROOF IN GEOMETRY CHỨNG MINH TRONG HÌNH HỌC IN BY A I FETISOV

schematically written in the form: "If A is B, C is D." For example: "If a quadrilateral is circumscribed about a circle, the sums of its opposite sides will be equal." The first part of the sentence, "If A is B", is termed the condition of the theorem, and the second, "C is D", is termed its conclusion. When the converse theorem is derived from the direct one the conclusion and the condi- tion change places. In many cases the conditional form of a theorem is more customary than the form "All S are P" which is termed the "categoric" form. However, it may easily be seen that the difference is inessential and that every conditional reasoning may easily be transformed into the categoric one, and vice versa. For example, the theorem expressed in the conditional form "If two parallel lines are intersected by a third line, the alternate interior angles will be equal" may be expressed in the categoric form: "Parallel lines intersected by a third line form equal alternate interior angles." Hence, our reasoning remains true of the theorems expressed in the conditional form, as well. Here, too, the simultaneous validity of the direct and the converse theorem is due to the fact that the classes of the respective concepts coincide. Thus, in the example considered above both the direct and the converse theorem hold,

BÁO CÁO TOÁN HỌC PRODUCT AND PUZZLE FORMULAE FOR GLN BELKALE KUMAR COEFFICIENTS POTX

S -deflated. Proof. It is slightly easier to discuss the case S c = { s } , and obtain the general case by shrinking one number s at a time. Let t ∈ [ 0, 1 ] , and change the puzzle regions as follows: keep the angles the same, but shrink any edge with label s to have length t . (This wouldn’t be possible if e.g. we had triangles with labels s, s, j 6 = s , but we don’t.) For t = 1 this is the original BK-puzzle P , and for all t the resulting total shape is a triangle. Consider now the BK-puzzle at t = 0 : all the s -edges have collapsed, and each ( i, s )- or ( s, i )-region has shrunk to an interval, joining two i -regions together.

REAL ANALYSIS WITH ECONOMIC APPLICATIONS - CONTENT PREFACE PPS

geometric (non)linear analysis.) However, because they play only a minor role in modern economic theory, I do not at all discuss topics like Fourier analysis, Hilbert spaces and spectral theory in this book. While I assume here that the student is familiar with the notion of “proof” — within the fi rst semester of a graduate economics program, this goal must be achieved — I also spend quite a bit of time to tell the reader why things are proved the way they are, especially in the earlier part of each chapter. At various points there are (hopefully) visible attempts to help one “see” a theorem (either by discussing informally the “plan of attack,” or by providing a “false-proof”) in addition to con fi rming its validity by means of a formal proof. Moreover, whenever it was possible, I have tried to avoid the rabbit-out-of-the-hat proofs, and rather gave rigorous arguments which “explain” the situation that is being analyzed. Longer proofs are thus often accompanied by footnotes that describe the basic ideas in more heuristic terms, reminiscent of how one would “teach” the proof in the classroom. 1 This way the text is hopefully brought down to a level which would be readable for most second or third semester graduate students in economics and advanced undergraduates in mathematics, while it still preserves the aura of a serious analysis course. Having said this, however, I should note that the exposition gets less restrained towards the end of each chapter, and the analysis is presented without being overly pedantic. This goes especially for the “starred” sections which cover more advanced material than the rest of the text.

BÁO CÁO TOÁN HỌC: "THE HAMILTONIAN P–MEDIAN PROBLEM" PPSX

In this paper, we will present a polyhedral representation of the Hamiltonian p - median polytope by inequalities which avoid the variables y i k . For certain applications it is necessary to permit loops, i.e. circuits which consist of only one point and whose arc set is { ( i, i ) } . Such a single depot supplies itself and no other customers. But there are also examples where it makes no sense to permit single loops. In such a case, each circuit must contain at least two different vertices. The costs for loops can be viewed as the costs for distributing the goods (transporting the people) within depot i . In this paper, we will discuss only the situation where loops are not allowed. It is not difficult to transform our results into the more general case.

LAPLACE TRANSFORMS THEORY PROBLEMS AND SOLUTIONS

Theorem 41.2 Existence Suppose that ft is piecewise continuous on t ≥ 0 and has an exponential order at infinity with |ft| ≤M eat for t ≥C.Then the Laplace transform Fs = Z ∞ 0 fte−stdt [r]

BÁO CÁO HÓA HỌC RESEARCH ARTICLE GENERALIZATIONS OF THE NASH EQUILIBRIUM THEOREM IN THE KKM THEORY POTX

In this paper, we study several stages of such developments from the KKM principle to the Nash theorem and related results within the frame of the KKM theory of abstract convex spaces. In fact, we clearly show that a sequence of statements from the partial KKM principle to the Nash equilibria can be obtained for any space satisfying the partial KKM principle. This unifies previously known several proper examples of such sequences for particular types of KKM spaces. More precisely, our aim in this paper is to obtain generalized forms of the KKM space versions of known results due to von Neumann, Sion, Nash, Fan, Ma, and many followers. These results are mainly obtained by 1 fixed point method, 2 continuous selection method, or 3 the KKM method. In this paper, we follow method 3 and will compare our results to corresponding ones already obtained by method 2 .

Đề tài " The density of discriminants of quartic rings and fields " doc

while, the cubic analogue of Theorem 4 may be obtained by combining the work of Davenport-Heilbronn [15] with that of Cohn [10]. An important ingredient that allows us to extend the above cubic results to the quartic case is a parametrization of quartic orders by means of two in- tegral ternary quadratic forms up to the action of GL 2 ( Z ) SL 3 ( Z ), which we established in [3]. The proofs of Theorems 1–5 thus reduce to counting integer points in certain 12-dimensional fundamental regions. We carry out this count- ing in a hands-on manner similar to that of Davenport [13], although another crucial ingredient in our work is a new averaging method which allows us to deal more eﬃciently with points in the cusps of these fundamental regions. The necessary point-counting is accomplished in Section 2. This counting result, together with the results of [3], immediately yields the asymptotic density of discriminants of pairs ( Q, R ), where Q is an order in an S 4 -quartic ﬁeld and R

Independent And Stationary Sequences Of Random Variables - Chapter 7 potx

166 RICHTER'S LOCAL THEOREMS ; BERNSTEIN'S INEQUALITY Chap . 7 „ 3 . Calculation of the integral near a sađle point From now on B will denote a bounded quantity, not necessarily the same from one occurrence to another . If Itf <n - I (log n) 2 ,

BÁO CÁO TOÁN HỌC EAR DECOMPOSITIONS IN COMBED GRAPHS PPS

T . Consequently G is conservative with respect to a minimum t -join. Suppose now that ( G, t ) is a join covered pair, where G is connected and bipartite with bipartition ( A, B ), and that t ( b ) = 1 for each b ∈ B . Then ( G, t ) is said to be a basic comb if τ ( G, t ) = |B| . This equation holds if and only if there is a t -join T , necessarily a minimum one, such that each vertex in B is incident with a unique edge of T . Moreover, since the pair ( G, t ) is join covered, every edge of G must belong to such a t -join. Fix such a minimum t -join T and, if possible, choose an edge e / ∈ T . Then e belongs to another minimum T -join T 0 . Furthermore, T ⊕ T 0 is a cycle and therefore a union of disjoint circuits, one of which contains e . Since each vertex of B is incident with just one edge of each of T and T 0 , this circuit must be balanced. Thus e belongs to a balanced circuit. If G is 2-connected, then the same must be true even if e ∈ T : if v is the end of e in

BÁO CÁO TOÁN HỌC SHORT CYCLES IN DIGRAPHS WITH LOCAL AVERAGE OUTDEGREE AT LEAST TWO PPTX

3 Conclusion and Open Problem The Caccetta-H¨ aggkvist Conjecture predicted a very interesting relationship among vari- ous fundamental parameters of digraphs: the girth, the degree and the number of vertices. It has been studied without general resolution since its appearance in 1978. Lacking ap- propriate methods to prove it, people tend to consider more general problems, in which the minimum outdegree condition δ + ( G ) ≥ r is relaxed so that it may be easier to em- ploy induction and some other proof techniques. Given the above strategy, one could expect that the most difficult part is to find an appropriate ‘generalized statement’. This has led to a number of stronger conjectures (see [3], for example). Motivated by Corol- lary 3, we present the following conjecture which is stronger than the Caccetta-H¨ aggkvist Conjecture.

BÁO CÁO TOÁN HỌC: "ON THE LIVINGSTONE-WAGNER THEOREM" POT

Main Theorem Let ( G, Ω) be a permutation group and let S be a family of subsets of Ω . If s < t are integers with s + t ≤ m( S ) then n s ( G, S ) ≤ n t ( G, S ) . The purpose of this note is to bring together various results in combinatorics which are all linked to each other via this theorem. In the first instance we should mention the theorem of Livingstone and Wagner [13] on the orbits of permutation groups when acting on subsets: this is the particular case when S = { Ω } . There are however many other applications of the theorem in combinatorial topology, graph theory and other parts of combinatorics which are new or simplify existing proofs. These applications will be stated first. In Section 2 we prove a more general orbit theorem on automorphism groups of partially ordered sets which contains the Main Theorem as a special case. There we also provide the proofs of the corollaries. The most interesting open question is: What else can be said about the sequence

GIẢI TÍCH HÀM

When I was a student, the only text on functional analysis was BanachỢs original classic, wriften im 1932; Hille's book appeared in time to serve as my graduation present. For Hilbert space there was StoneỢs Colloquium publication, also from 1932, and Ếz.-Nagy's Ergebnisse volume. Since then, our cup hath run over; frst came Riesz and Ếz.-Nagy, then Dunford and Schwartz, Yosida, later Reed and Simon, and Rudin. For Hilbert space, there was Halmos's elegant slender volume, and Achiezer and Glazman, all of which I read with pleasure and profit. Many, many more good texts have appeared since. Yet I believe that my book offers something new: the order in which the material 1s arranged, the interspersing of chapters on theory with chapters on applications, so that cold abstractions are made flesh and blood, and the inclusion of a very rich fare of mathematical problems that can be clarified and solved from the functional analytic point of view.

ELECTRIC CIRCUITS 9TH EDITION P49 DOCX

Observe that the right-hand side of Eq. 12.96 may be written as lim / -j-e"dt + / -^e~ st dt . * - ° ° \ Mr dt J it dt J As s —* co, (df/dt)e~ st —> 0; hence the second integral vanishes in the limit. The first integral reduces to / ( 0+ ) - /(0~), which is independent of s. Thus

BÁO CÁO TOÁN HỌC GENERATING FUNCTIONS ATTACHED TO SOME INFINITE MATRICES PPSX

Our proof of Theorem I is easier than Gessel’s proof of his special case of the corollary. The reason for this is that by working over Λ rather than over F we are able to restrict our study to walks with step-sizes in {− 1 , 0 , 1 } . (A complication, fortunately minor, is that the weights must be taken in the non-commutative ring Λ.) Our proof is well-adapted to finding an explicit polynomial relation between G ( V ) and z ; we’ll work out a few examples. This paper would not have been possible without Ira Gessel’s input. I thank him for showing me tools of the combinatorial trade.

ADAPTIVE CONTROL 2011 PART 12 PPTX

National Pingtung University of Science and Technology Pingtung, Taiwan 1. Introduction For low cost, easy assembly and less maintenance, overhead crane systems have been widely used for material transportation in many industrial applications. Due to the requirements of high positioning accuracy, small swing angle, short transportation time, and high safety, both motion and stabilization control for an overhead crane system becomes an interesting issue in the field of control technology development. Since the overhead crane system is underactuated with respect to the sway motion, it is very difficult to operate an overhead traveling crane automatically in a desired manner. In general, human drivers, often assisted by automatic anti-sway system, are always involved in the operation of overhead crane systems, and the resulting performance, in terms of swiftness and safety, heavily depends on their experience and capability. For this reason, a growing interest is arising about the design of automatic control systems for overhead cranes. However, severely nonlinear dynamic properties as well as lack of actual control input for the sway motion might bring about undesired significant sway oscillations, especially at take-off and arrival phases. In addition, these undesirable phenomena would also make the conventional control strategies fail to achieve the goal. Hence, the overhead crane systems belong to the category of incomplete control system, which only allow a limited number of inputs to control more outputs. In such a case, the uncontrollable oscillations might cause severe stability and safety problems, and would strongly constrain the operation efficiency as well as the application domain. Furthermore, an overhead crane system may experience a range of parameter variations under different loading condition. Therefore, a robust and delicate controller, which is able to diminish these unfavorable sway and uncertainties, needs to be developed not only to enhance both efficiency and safety, but to make the system more applicable to other engineering scopes.

BÁO CÁO HÓA HỌC:" RESEARCH ARTICLE GREEN’S FUNCTION FOR DISCRETE SECOND-ORDER PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS ˇ SVETLANA ROMAN AND ARTURAS STIKONAS ¯" PPTX

In this paper, expressions of Green’s functions for 1.1 have been obtained using the method of variation of parameters 12 . The advantage of this method is that it is possible to construct the Green’s function for a nonhomogeneous equation 1.1 with the variable coe ﬃ cients a 2 , a 1 , a 0 and various additional conditions e.g., NBCs . The main result of this paper is formulated in Theorem 4.1 , Lemma 5.3 , and Theorem 5.4 . Theorem 4.1 can be used to get the solution of an equation with a di ﬀ erence operator with any two linearly independent additional conditions if the general solution of a homogeneous equation is known. Theorem 5.4 gives an expression for Green’s function and allows us to find Green’s function for an equation with two additional conditions if we know Green’s function for the same equation but with di ﬀ erent additional conditions. Lemma 5.3 is a partial case of this theorem if we know the special Green’s function for the problem with discrete initial conditions. We apply these results to BVPs with NBCs: first, we construct the Green’s function for classical BCs, then we can construct Green’s function for a problem with NBCs directly Lemma 5.3 or via Green’s function for a classical problem Theorem 5.4 . Conditions for the existence of Green’s function were found. The results of this paper can be used for the investigation of quasilinear problems, conditions for positiveness of Green’s functions, and solutions with various BCs, for example, NBCs.

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