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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

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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]

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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.

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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.

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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.

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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 .

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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

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Theorem 4.2 The inverse

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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

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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.

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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]

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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

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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.

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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

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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.

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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]

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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** .

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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.

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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,

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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.

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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.

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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.

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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]

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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 .

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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

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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 ,

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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.

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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

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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.

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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

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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.

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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.

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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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Từ khóa: nuclear energy reactions advantages and disadvantageslow energy nuclear reactions the realism and the outlookpower and energy in circuits mastering physics± pss 25 1 power and energy in circuits mastering physicspss 25 1 power and energy in circuits mastering physicsNghiên cứu sự biến đổi một số cytokin ở bệnh nhân xơ cứng bì hệ thốngNghiên cứu tổ chức chạy tàu hàng cố định theo thời gian trên đường sắt việt namđề thi thử THPTQG 2019 toán THPT chuyên thái bình lần 2 có lời giảiGiáo án Sinh học 11 bài 13: Thực hành phát hiện diệp lục và carôtenôitGiáo án Sinh học 11 bài 13: Thực hành phát hiện diệp lục và carôtenôitGiáo án Sinh học 11 bài 13: Thực hành phát hiện diệp lục và carôtenôitPhát triển mạng lưới kinh doanh nước sạch tại công ty TNHH một thành viên kinh doanh nước sạch quảng ninhPhát triển du lịch bền vững trên cơ sở bảo vệ môi trường tự nhiên vịnh hạ longPhát hiện xâm nhập dựa trên thuật toán k meansNghiên cứu, xây dựng phần mềm smartscan và ứng dụng trong bảo vệ mạng máy tính chuyên dùngTìm hiểu công cụ đánh giá hệ thống đảm bảo an toàn hệ thống thông tinThơ nôm tứ tuyệt trào phúng hồ xuân hươngKiểm sát việc giải quyết tố giác, tin báo về tội phạm và kiến nghị khởi tố theo pháp luật tố tụng hình sự Việt Nam từ thực tiễn tỉnh Bình Định (Luận văn thạc sĩ)Tranh tụng tại phiên tòa hình sự sơ thẩm theo pháp luật tố tụng hình sự Việt Nam từ thực tiễn xét xử của các Tòa án quân sự Quân khu (Luận văn thạc sĩ)Nguyên tắc phân hóa trách nhiệm hình sự đối với người dưới 18 tuổi phạm tội trong pháp luật hình sự Việt Nam (Luận văn thạc sĩ)Giáo án Sinh học 11 bài 14: Thực hành phát hiện hô hấp ở thực vậtGiáo án Sinh học 11 bài 14: Thực hành phát hiện hô hấp ở thực vậtChiến lược marketing tại ngân hàng Agribank chi nhánh Sài Gòn từ 2013-2015Đổi mới quản lý tài chính trong hoạt động khoa học xã hội trường hợp viện hàn lâm khoa học xã hội việt namMÔN TRUYỀN THÔNG MARKETING TÍCH HỢP