Showing papers on "Divisor published in 1981"

Counting divisors with prescribed singularities

Abstract: Given a family of divisors {/),} in a family of smooth varieties { Ys} and a sequence of integers mx, . . . , m,, we study the scheme parametrizing the points (s,yx, . . . ,y,) such that^, is a (possibly infinitely near) m,-fold point of D,. We obtain a general formula which yields, as special cases, the formula of de Jonquieres and other classical results of Enumerative Geometry. We also study the questions of finiteness and the multiplicities of the solutions.

The arithmetic mean of the divisors of an integer

Abstract: Dedicated to Emil Grosswald on the occasion of his sixty-eighth birthday .

Some problems of analytic number theory III

Abstract: The main theme of this paper is to systematize the Hardy-Landau $\Omega$ results and the Hardy $\Omega_{\pm}$ results on the divisor problem and the circle problem. The method of ours is general enough to include the abelian group problem and the results of Richert and the later modifications by Warlimont, and in fact theorem 6 of ours is an improvement of their results. All our results are effective as in our earlier paper II with the same title. Some of our results are new.

A class of primality criteria formulated in terms of the divisibility of binomial coefficients

Abstract: We find a class of theorems of the type “q is a prime number iff R(q) is a divisor of the binomial coefficient\(\left( {\begin{array}{*{20}c} {S(q)} \\ {T(q)} \\ \end{array} } \right)\) ”; here R, S, T are certain fully significant functions that are superpositions of addition, subtraction, multiplication, division, and raising to a power. Similar criteria were also obtained for prime Mersenne numbers, prime Fermat numbers, and twin-prime numbers.

A rational arithmetic processor

Abstract: An arithmetic processor based upon a rational representation scheme is examined. The key feature of this rational processor is its ability to efficiently reduce a result ratio to its irreducible form (the greatest common divisor of the numerator and denominator is unity). The reduction algorithm presented generates the reduced ratio in parallel with the evaluation of the ratio's greatest common divisor. Hardware designs for the reduction algorithm and the basic arithmetic operations are given.

Rates of decrease of sequences of powers in commutative radical Banach algebras.

Abstract: Every element 6 of a radical Banach algebra & satisfies l iπv^ ||6||=0. We are concerned here with the existence of a lower bound for the rate of decrease of the sequence (||δ||) under various assumptions over b and ^ , when b is not nilpotent and & commutative. If the nilpotents are dense in & then for every sequence (λn) of positive reals there exists a nonnilpotent b e & such that lim inf*-«, ||6||/ϋ»=0. A stronger result holds if & possesses furthermore a bounded approximate identity. On the other direction if & has no nilpotent element and if some element of & which is not a divisor of zero acts compactly on & then there exists a sequence (2n) of positive reals such that \immfn-+0O\\b n\\lλn— + °o for every nonzero be&. Also there exists universal lower bounds for the rate of decrease of ll

Pulse repetitive frequency discrimination system

Abstract: PURPOSE:To detect and separate a string of constant-period pulses from a string of mixed pulses by considering a time interval between two pulses, belonging to mixed pulses, to be a divisor and by performing processing by dividing times from one end pulse to respective pulses by it in sequence CONSTITUTION:A mixed pulse string from terminal 1 is applied via pulse-interval integrating circuit 2 to memory circuit 3 and time intervals from the 1st pulse of the pulse string to respective pulses are stored Among data from circuit 3, the maximum pulse interval is detected 4 and sent to divisor selecting circuit 5 Circuit 5 selects and sends this to dividing circuit 6, which divides by this interval the time intervals up to respective pulses read out from circuit 3, and from the results, a divisibility frequency is detected 7 and sent to divisibility frequency deciding circuit 8 and dividing circuit 6 Once circuit 8 decides that the divisibility frequency does not reach a fixed number, circuit 5 selects a new divisor to repeat the division and when the divisibility reaches the fixed number, a pulse interval selected at that time as the divisor is output as a repetitive frequency, so that circuit 6 removes the pulse string from circuit 3

Frequency generator for generating signalling frequencies in a multifrequency telephone apparatus.

Abstract: of EP00212961. Frequency generator for generating signalling frequencies in a multifrequency telephone apparatus comprising a quartz oscillator (1) ; a first frequency divider (2) which is connected to the quartz oscillator (1) and has a fixed divisor ; a frequency divider (3, 4) which is connected to the first frequency divider (2) and has an integral divisor which can be set by a programming network (7) for generating frequencies selected through the programming network (7) ; and a digital/analog converter comprising a fourth frequency divider (5) with a fixed divisor, and a weight network (6) connected to the steps of the fourth frequency divider (5) for forming step-shaped signals of approximate sine wavefrom ; characterized in that the frequency divider (3, 4) with a settable integral divisor comprises and AND member (37) and an auxiliary flip-flop (4) as last step ; in that the output of the flip-flop (4h via 43h) is connected to a first input of the AND member (37) ; in that at least one further input of the AND member (37) is connected to the output of a divider step (31, 32) preceding the auxiliary flip-flop (4h) ; and in that the output of the AND member (37 via 341) is connected to the reset input of the auxiliary flip-flop (4h), and, controlled by the programming network (7 via 731, 732, 741, ..., 74p), to the set or reset input of at least one divider step (31, 32, 41, ..., 4p) preceding the auxiliary flip-flop (4h).

On Class Numbers of Hyperelliptic Function Fields-3-

Abstract: Let F =GF (p) be a finite prime field of characteristic p≠2.Let K=F (x, y) be an algebraic functicon field over F defined by an equation y2 = xn-a (a≠0,a∈F), where n means an odd number so that n>1 and p?n. Let h be the class number of K and g the genus of K. Then, it is obvious that h=p+1 if n=3 and p≡2 mod 3. This particular fact can be generally expressed as follows; Given n, there exists an integer c such that h = (p + 1) g whenever p≡c mod n. In this note, it is shown that this generalization is true in the particular case of n = 5 and of n=7. 1.Jntroduction. Let F=GF(P)be、a finite prime field of characteristic p≠2. Let n be an odd number satisfying n>l and pyn. Throughout this note,K=F (x,y)means an algebraic function field over F defined by an equation y2=Xn一一a (a≠0,a∈F).If we denoteby g the genus of K,itis obvious that g=(n-1)/2. Let h be the class nllmber of K,i・e▲,the order of the finite group of divisor classes of degree zero. We willthen discuss the followlng queStion; Does there exist aninteger cwhich depends only on n such that h=(P+1)g whenever P≡C mOd n? In the case of n=3,We had an answerin the affirmative.([5],Theoreml (i)). When h=Pg+l then the similar q11eStion was discussed.([6]).In this note,We Wish to give an answer onlyin the case of n=5and of n=7.In doing so,We Will recall a method of estimating class numbers of algebraic function fields without proofs but with references. Let L(u)=1+alu+a2u2+...+Pg-2a2u2g 2+Pg lalu2g-1+Pgu2g be the L-function of K.Thenitis obvious that h=L(1). Asis well known,the explicit expression for coefficients al,a2and a3Can be putin the form 2 Tadashi NASHIO a, = Nl ~ ( p + I ) (1) 2a2 = N12 ~ (2 p + I ) N1 + 2N2 + 2 p 6a3 = N,a _ 3 pN12 + (3 p l) N1 ~ 6 ( p + I ) N2 + 6NIN2 + 6N3 where Nd means the number of prime divisors of degree d of K. (M.L.Madan and C.S.Queen [2], p. 427). Thus , for our present purpose , it is enough to compute Nd . Since Nd depends on the number of prime divisors of degree one in some constant field extensions of K, in S 2 we will investigate the number N(Kd) of prime divisors of degree one of an algebraic function field Kd Over a finite field Fd . In S 3 , we will compute h in the case of n=5 and of n=7. 2. The number of prime divisors of degree one. Let Kd be the constant field extension of K of degree d and let Fd be the finite field GF (pd) with pd elements . Let us denote by N(Kd) the number of prime divisors of degree one of Kd . We will then consider N(Kd) under the assumption p~~-1 mod n . THEOREM l. If d~;1 mod 2 and p~~-1 mod n, then the equality N(Kd) =pd+1 holds . PROOF. By the definition of N(Kd) , we have N(Kd) =1+# {(a, p) EFdXFd ; p2=a a} So we need to estimate the last term in this formula . Since our assumtions d~~I mod 2 and p~~-1 mod n lead to (pd-1,n) =1, we can get Fdn=Fd in view of the fact that F*=Fd{ O } is a cyclic group of order pd-1 . This implies that #{(a,O)EFdXFd ; a"=a} =1 and #{(a,p)EFdXFd ; p~0 , p2=cr"-a} =pd-1. Therefore we have N(Kd) =1 + l+pd -1 = pd +1 . We will now consider N(Kd) in the cas9 of d=2. THEOREM 2. If p~E-1 mod n and n=2g+1, then N(K2)=p'+2gp+1 holds. We will prepare some lemmas for the proof of this theorem. The following lemma will be proved on the basis of the properties of Hasse-Witt matrices of the algebraic function field Kd Over Fd . LEMMA I . If p~~ I mod n, then N (Kd) ~= I mod p for an arbitrary positive integer d . PROOF. For O~u,v~;g I , Iet A~,, be the coefficient of x"+* in the following pol ynomial I~((x"-a) (p~')/2x"'1)=1~( ~] (p~1)/2 (p-1)/2 (_a) (p~1) 2 'x"""'t) where I~ means ( ) .=0 r the p~1 Iinear operator satisfying r x"/p if p j w I~ (x") = The matrix A=(A~,.) is called the Hasse-Witt matrix. (L.Miller [3]) . Since it is easy in our case to show that nr+u+1~p(v+1) for every O~u, v~g-1, we On Class Numbers of Hyperelliptic Function Fields,五 3 haveん,ひニO i.e.,メ4=O. Therefore the desired assertion.N(K4)...1mod p follows at once from z4=O.([4],Theorem). LEMMA2.1∫ρ......一1modη,診h6πN(K2)......O mod2αη42〉(K2)...3modηhol4。 PROOF. Since(ρ一1,π)=1,we get#{α∈F1=GF(p)1απ=α1=1. This lead to #{α∈F2;α露=α/-n,because F2contains,in our case,anπth primitive root of unity. Moreover,it is clear that#{(α,β)∈F2×F21β2一αη一α,β≠Ol≡O mod2。Hence we get the first part of the lemma as follows. N(K2)=1十#{(α,O)∈172×F2;αη=8} +#{(α,β)∈F2×F2;β≠0,β』α”一α}≡...1+η≡O mod2。 We will now prove the second assertion.Since F2conta三ns anπth primitive root of unity,we have #{(α,β)∈F2×F、;αη=β2+α,α≠0}...≡0:modπ. Therefore,because of#{β∈F2;β2......一α}=2,we get N(K2)=1+#/(O,β)∈F2×F21β2・=一α} +#/(α,β)∈F2×F2;α≠O,β2一α”一‘zl...≡1+2...3mo(iη. This completes the proof of the lemma. Now let us tum to the proof of Theorem2. PROOF of Theorem2。As is well known,the inequalitiesρ2十1-29ρ≦N(K2)≦ρ2 十1十29ρhold.(M。Eichler[1],P.306). Therefore Lemma l and the first part of Lemma21ead to 2V(K2)一ρ2+1+卿(形一〇,±2,±4,......,±29). Using the second part of Lemma2,we haveρ2十1十郷ρ......3modη. Therefore we can easily get解=2g because of our assumptionsρ≡...一l modπand n=29十1. Hence we have our assertion1〉(K2)ニρ2十1十2gρ. 3. Results. Let us now consider the question in §10nly in the case ofη=5 andπ=7.In fact,we can answer our question in the case ofπ=5in the affirmative as follows. THEOREM3.L8君F・=GF(ρ)68α方競8ρプ麹8方8140∫6肋7疏爾5漉ρ≠2.L8オ K-F(諾,ッ)68αhッPθプ81妙痂血n6オプ・n方614・η6プ.F48伽8づ6ツαn69微漉・ηプー諾5-8 (α≠0,α∈F).Z)8π・」のッhオh召61α55π蹴み8プげK。1∫ρ≡4mod5,診h8nh一(ρ+1)2 例4L(%)一1+2勿2+ρ2π4. PROOF.ApPlying Theorem l to4=1we haveノ〉1=N(K1);ρ十1. Moreover, apPlying Theorem2to9=2,we have1〉(K2)=ρ2十4ρ十1. So we get N2=ρ(ρ十3)/2 in view of the fact that the relation among1〉1,ハ72and N(K2)is given,by N(K2) ・=2〉、十21〉2. Therefore,by making use of the formula(1),we can easily obtain ごz1=0,ζz2=・2ρ, L(%)=1十2ρ%2十ρ2π4and h=L(1)=(メ》十1)2. Finally we will give an affirmative answer to our question in the case ofπ=7. THEOREM4.LαF=GF(カ)68α五n漉ρ7伽8方8140∫6hαプ観副5痂ρ≠2,Lα K=F(‘τ,‘y) 68αh‘yメ》8プ6114》擁c ∫診¢π6擁on 方614 0ηθプ F 46Lβn846ツ‘zn 6gz¢‘z擁on ツ2=‘じ7一α (α≠0,召∈F).Z)eπ・詑δッ勉h861α∫∫n%〃z6eプ・∫K.1∫ρ・三6mod7,彦h8nん一(ρ+1)3 4 Tadashi ~YVASHIO and L(u) = I + 3pu2 + 3p2u' + p3u6 . PROOF . As applications of Theorem I to d=1 and d=3, we have N*= N(K*) = p+1 and N(K3) =p3+1 . Consequently the formula N(K3) = N* + 3N3 Ieads to N*= (p3-p)/3 . Similarly , applying Theorem 2 to g = 3 , we have N(K,) = p' + 6p + I . Therefore the formula N(K2) =N*+2N2 also leads to N2= (p2+5p)/2. Hence, by means of the formula (1), rt Is easy to check on al=a3=0, aa=3p, L(u) =1+3pu2+3p'u'+p3u' and h=L(1) =(p+1)". This completes the proof of the theorem .

Representation of an integer as a prime plus a product of two small factors

Abstract: In this paper we consider the following problem, which seems to have been brought to light fairly recently by M. Car. Can every sufficiently large integer n be expressed as n = p + ab with p prime and 1 ≤ a , b ≤ n ½ ? Certainly one should expect this to be possible. Taking b = 1, for example, p will be restricted to the range n − n ½ ≤ p n , and this interval is conjectured to contain a prime, for large enough n . Alternatively, providing that n is not a square, we expect n = p + a 2 to be solvable for sufficiently large n . However, although the statement that n = p + ab , with a , b ≤ n ½ , is far weaker than either of the aforementioned conjectures, it is nevertheless rather tricky to show that solutions must in fact exist.

Complex hill's equation, complex korteweg - de vries flows and jacobi varieties

Abstract: Let q be a point in the phase space (DIAGRAM, TABLE OR GRAPHIC OMITTEDPLEASE SEE DAI) of complex-valued infinitely differentiable functions of a real parameter, of period 1 Then the spectrum of the Hill's operator (DIAGRAM, TABLE OR GRAPHIC OMITTEDPLEASE SEE DAI) in a class of functions of period 2 is a countable set in the complex plane, we assume that 2g + 1 < (INFIN) of these (lamda)(,0), (lamda)(,1), , (lamda)(,2g) are simple the infinite compliment being double, with equal spectral multiplicities The Q is the isospectral class of smooth potentials having a meromorphic extension into the whole complex plane We prove the existence of Q and construct it by non-linear flows in (DIAGRAM, TABLE OR GRAPHIC OMITTEDPLEASE SEE DAI) determined by the complex Korteweg-de Vries equation (DIAGRAM, TABLE OR GRAPHIC OMITTEDPLEASE SEE DAI) and the complex higher order KdV flows {13}, {22}, and {29} These flows are completely integrable and proceed along straight lines with constant speed in the Jacoby variety of the hyperelliptic curve (DIAGRAM, TABLE OR GRAPHIC OMITTEDPLEASE SEE DAI) determined by the simple spectrum Q is recovered from J by the Its-Matveev formula {20} and the role of the theta divisor (theta), which is an irreducible singular subvariety of J of codimension 1, is clarified, in particular some KdV flows proceed in O An auxiliary (Direchlet) spectrum gives rise to natural coordinates for Q: an auxiliary divisor on K, which is a point in S('g)(K): the g-fold symmetric product of the curve The KdV flows in S('g)(K) are globalized by desingularizing the equations regulating them, using concepts from algebraic function theory This leads to a natural extension of Q to singular potentials, having poles of their meromorphic extension as singularities We completely characterize Q and its extension as being isomorphic to the Jacobi variety of K

Divisor conversion type highhspeed division system

Abstract: PURPOSE:To realize a high-speed division with an extremely small error for the system in which a division is carried out via a high-speed multiplying device, by performing the conversion of the binary divisor and a division with no successive repetition by the division of the conversion number. CONSTITUTION:The system comprises with the fundamental constitution using the divisor conversion read-only memories 4-6, the multiplying devices 1-3 plus adder rows 7-7; and these component units are coupled in a single direction to secure an asynchronous working. In this case, the conversion number of the Nh digits corresponding to the divisor of the h digits in 1:1 is used and then grouped into the division numbers in the order of the higher-rank digits, and then three units of the division number are delivered from the memories 4-6. This output is suppllied to one side of the devices 1-3, and the dividend is supplied to the other side of the devices 1-3 each to obtain the output of the 2h digits at the output. This output is grouped again into the division product of the upper and lower digits, and the upper and lower division products of the two different multiplying devices to form a facies are applied to the row of adders 8-10 to which the carry output/input line is connected. Thus a division is carried out with the conversion of the binary divisor with no successive repetition by the division of the conversion number. Then a high-speed division is possible with an extremely small error.

On homeomorphisms preserving principal divisors

Abstract: ABSTRACr. Let SI and S2 be compact Riemann surfaces of genus g > 1. Let T: SI S2 be a continuous map. The map T induces a group homomorphism from the group of divisors on S, into the group of divisors on S2* This group homomorphism will be denoted by the same letter T throughout this paper. If D = I mjp is a divisor on SI, then T(D)= XnI miT(p). If T is a holomorphic or an anti-holomorphic homeomorphism, then T(D) is a principal divisor on S2 if D is a principal divisor on SI. To what extent is the converse of this statement true? The answer to this question is provided by Theorem I of this paper: If T(D) is a principal divisor on S2 whenever D is a principal divisor on SI, then T is either a holomorphic or an anti-holomorphic homeomorphism.