fermat s theorem and euler s criterion

... and I ! = 0, this doesn’t really tell us anything useful about I Thus, P P we consider subsets of , rather than individual elements of e Distribution of B T  If is constant, then n o Z ... is a martingale under IP 192 Lemma 1.54 (Baye s Rule) If X is F t-measurable and  s  t  T , then f IE X jF s = Z s IE XZ tjF s : Proof: It is clear that Z 1s IE XZ tjF s is ... y2 dy (Substitute y = T + b) y = T + b = p 2T ,1 = 0: CHAPTER 17 Girsanov s theorem and the risk-neutral measure 191 fe We can also see that I B T  = by arguing directly from the density formula...

Ngày tải lên: 18/10/2013, 03:20

... (1) that (3) holds in the case that f is the characteristic function of an open subinterval I, and the same reasoning shows that this also holds if I is a closed or half-open subinterval Now, ... (Gowers s Theorem) There exists a positive constant ck such that any subset A in [1, N ] N with |A| (log log N )ck contains a non-trivial k−term artihmetic progression References [1] K Chandrasekharan, ... and hence so are the residues of 2ha (mod q) since (a, q) = Thus 2ha is congruent to each of 0,1,-1 (mod q) at most once, and the total contribution to S in these cases is therefore at most 3N ...

Ngày tải lên: 01/06/2014, 21:36

... Browder s Theorems and Some Results in IFNS In this section, we discuss the idea of fuzzy type of some Browder s fixed point theorems in intuitionistic fuzzy normed space by 26, 27 As a consequence ... Nonlinear Science and Its Applications, vol 3, no 2, pp 96–109, 2010 13 R Saadati, S Sedghi, and N Shobe, “Modified intuitionistic fuzzy metric spaces and some fixed point theorems,” Chaos, Solitons and ... ∗, ♦ is said to be intuitionistic fuzzy normed spaces for short, IFNS if X is a vector space, ∗ is a continuous t-norm, ♦ is a continuous t-conorm, and μ, υ are fuzzy sets on X × 0, ∞ satisfying...

Ngày tải lên: 21/06/2014, 11:20

... Terms on the RHS can be grouped together according to the types of X, so we can write p (S0 ) = c (S0 , S) S S0 G = c (S0 , S0 ) S0 G S + S S0 G c (S0 , S) S (40) Therefore, G S0 = c (S0 , S0 ) p (S0 ... induced by S ⊆ V G is the subgraph whose vertex set is S and whose edge set contains all the edges having both end vertices in S It is denoted by GS The subgraph of G induced by V G − S is denoted ... Remark It is is possible that for distinct j and k, the matrices N(Λj ) and N(Λk ) are equal In this case a multiplicity NIj is assigned to N(Λj ) and NIk is assigned to N(Λk ) while constructing...

Ngày tải lên: 07/08/2014, 13:21

... k-circular successions we can first choose m positions i1 , , im of k-circular successions among the first n − k ones and then construct a permutation π0 of order n−m without k-circular successions on ... k-circular succession of σ (i) Suppose sgnπ (σ(i)) = As σ(i) ∈ Tσ and σ −1 (σ(i)) = i, we have sgnπ (σ(i)) = sgnπ (i) · sgnπ (σ(i)) = sgnπ (i) (ii) Suppose that sgnπ (σ(i)) = sgnπ (i) As σ(i) ∈ Tσ and ... decomposition into blocks is 9|5 8|2| and we recover σ by putting parentheses around each block Lemma 15 For k ≥ 1, the mapping ϕ transforms the k-circular successions to k-linear successions and...

Ngày tải lên: 07/08/2014, 15:23

... this very successful study of the algebraic method behind Alon and Tarsi s Theorem, combinatorialists always search for purely combinatorial proofs, since this usually helps to understand the situation ... each such v ∈ V1P \ V1C Mrs Correct has to use (and use up) one eraser from the corresponding stack Sv She loses if she runs out of erasers and cannot that, i.e., if already Sv = ∅ for a still ... already have lost more erasers One example where Paint s winning strategy is like this is K3,3 with one eraser at each vertex Alon and Tarsi s Theorem In this section we discus a surprising connection...

Ngày tải lên: 08/08/2014, 11:20

... theory is not as well-developed as that of the first variation and the Euler- Lagrange equations We will consider these issues and several others in a geometric setting as suggested above, using various ... the first term on the right-hand side vanishes by our assumption on ξ (our second use of this assumption), so we must have = Θk ∧ (X ξ), and the induction hypothesis then gives X ξ = This is true ... differential system whose integral manifolds are graphs of Gauss maps of K = −1 surfaces in E3 In §4.3.4, we will discuss the B¨cklund transformation for this system, which a relates this particular...

Ngày tải lên: 31/03/2014, 15:56

... consider the remaining cases Separation of the sets considered in the remaining cases follows from Theorem 2.6 So, we must show that this separation is strict Thus we assume that the sets K0 and ... if X0 is a subspace of a Banach space X, if a set is closed relative to subspace X0 , it is closed with respect to X Theorem 2.7 Let X be a space as in Theorem 2.6, and let K0 and K1 be disjoint ... is easy to see that K0 ⊂ X0 , and that it has a nonempty interior relative to X0 We will consider all possible cases with respect to the position of the sets K0 and K1 , which are as follows:...

Ngày tải lên: 22/06/2014, 19:20

... remaining axioms are obviously ˜ ˜ satisfied by L Thus L satisfies the axioms of Theorem 1.1 Now suppose λ is a function from the self-maps of spaces in Ꮿ to the integers that satisfies the axioms We regard ... ∨ Sn ∨ · · · ∨ Sn , where Sn = Sn , then we have inclusions e j : Sn → k Sn into j j k the jth summand and projections p j : k Sn → Sn onto the jth summand, for j = 1, ,k j If f : k Sn → k Sn ... draft of this paper and giving us helpful suggestions References [1] [2] [3] [4] C Bowszyc, Fixed point theorems for the pairs of spaces, Bull Acad Polon Sci S r Sci Math e Astronom Phys 16 (1968),...

Ngày tải lên: 23/06/2014, 00:20

... this case, the map z → Cz is said to be transfer closed-valued by some authors Final Remarks (1) In most of our results, we showed that compact convex subsets of Hausdorff topological vector spaces ... topological vector spaces in some of Urai s results can be replaced by convex spaces with finite covers consisting of open (closed) neighborhoods of points of those spaces Urai s main tools are the partition ... argument on such covers and the Brouwer fixed point theorem This is why he needs Hausdorffness and compactness However, our method is based on a new Fan-Browder type theorem (Theorem 2.2), which is actually...

Ngày tải lên: 23/06/2014, 00:20

Ngày tải lên: 05/08/2014, 09:46

Ngày tải lên: 05/08/2014, 15:20

Ngày tải lên: 05/08/2014, 15:21

... then J belongs to C (W 1,p (Ω)) and satisfies conditions in classical Mountain Pass Theorem This situation has been studied in [8] p In this paper, we consider h ∈ L p−1 (Ω) In this case, the value ... linear subspace of E Suppose I belongs to Cw (E) and satisfies the Palais-Smale condition Assume the following conditions hold (i) There exist a bounded neighborhood D of in V and a constant α such ... Mountain-Pass Theorem, Saddle Point Theorem and the Z2 version of Mountain Pass Theorem were proved in [9] for functionals of class C (E) In the present paper, we extend these results to functionals of...

Ngày tải lên: 06/08/2014, 05:20

... of q-series identities, J Combin Theory Ser A, 99 (2002), 1–16 [6] G Gasper, Elementary derivations of summation and transformation formulas for q-series, In: Special Functions, q-Series and Related ... q-binomial theorem and Theorems 1.2 and 1.3 Our combinatorial proof of the q-binomial theorem is based on Theorem 2.1, and is essentially the same as that of Alladi [2] or Pak [8] Proof of Theorem 1.1 ... q)∞ which is symmetric in a and b Note that the special case z = of (1.2) has also appeared in the literature (see Stockhofe [9] and Pak [8, 2.2.4]) Rewrite the left-hand side of (1.3) as (z; q)n...

Ngày tải lên: 07/08/2014, 07:21

... this assertion for locally finite G and discuss a possible application to hamiltonicity problems Finally, in Section we discuss extensions of the above results to non-locally-finite graphs and ... In this section we are going to discuss generalisations of the results of this paper to non-locally-finite graphs and other spaces Let us start with an example The graph G of Figure consists of ... bears some similarity to the first singular homology group H1 of |G|, as both are based on circles, i.e closed 1-simplices, in |G| Diestel and Spr¨ ssel [16] investigated the precise relationship...

Ngày tải lên: 07/08/2014, 21:21

... the summation formulae in Lemmas and Perhaps more interesting is the result of interpreting the left sides of S1 6 and S9 4 using the summation formula in Lemma As is well known, the identity at S1 4 ... partitions and the result is proved Graphically, we may describe these transformations as follows In each case, we start with the usual Ferrers diagram of the partition It can be seen that the largest ... Preussischen Akadamie der Wissenschaften, Physikalisch-Mathematische Klasse (1917), 302-321 ¨ [27] A Selberg, Uber einige arithmetische Identit¨ten, Avrandlinger Norske Akad 8, a 1936 [28] A V Sills,...

Ngày tải lên: 08/08/2014, 14:22

... the same assumption, the number of hypersurfaces in our result is smaller than that in the all previous results on uniqueness of meromorphic mappings sharing hypersurfaces Also in the case of ... weights for the case of hypersurfaces in complex projective varieties In the last section of this paper, we prove a uniqueness theorem for meromorphic mappings sharing hypersurfaces in subgeneral ... assertion (v) is also satisfied The lemma is proved Second main theorems for hypersurfaces Let {Qi }i∈R be a set of hypersurfaces in Pn (C) of the common degree d Assume that each Qi is defined by...

Ngày tải lên: 16/10/2015, 14:14

... ideals in K Since these available summands are now power of prime numbers, they form a sparser set compared to the set of prime numbers and so their sum will span a sparser set relative to the set ... (N ) = As S( N ) log log N N s 1 + O N s 1 s+ 1 logs N log N , where As is a positive constant depending on s and K but independent of N , and S( N ) denotes the singular series If N ≡ s mod 2D, ... these tools is an inequality for exponential sums We shall also introduce some typical arithmetic functions, Ramanujan sums, infinite products and Euler products In the last section of this chapter,...

Ngày tải lên: 16/10/2015, 15:36

... remarkable properties: (i) The class of semi-algebraic sets is closed with respect to Boolean operators; a Cartesian product of semi-algebraic sets is a semi-algebraic set; (ii) The closure and the interior ... set D = {p ∈ P | Fp S} is open and dense in P Moreover, if P, X, Y, and S are semi-algebraic sets and if F is a semi-algebraic map, then D is also semi-algebraic Proof The proof of openness and ... ,n First of all, since the number of faces of Γ is finite and a finite intersection of open and dense semi-algebraic sets is open and dense semi-algebraic, it suffices to consider the case where...

Ngày tải lên: 04/09/2016, 09:08