scholarly journals Regular formal modules in local fields and irregularly degree

Author(s):  
Natalya K. Vlaskina ◽  
◽  
Sergei V. Vostokov ◽  
Petr N. Pital’ ◽  
Aleksey E. Tsybyshiev ◽  
...  

In this paper we investigate the irregular degree of finite not ramified local field extantions with respect to a polynomial formal group and in the multiplicative case. There was found necessary and sufficient conditions for the existence of primitive roots of ps power from 1 and (endomorphism [ps]Fm) in L-th unramified extension of the local field K (for all positive integer s). These conditions depend only on the ramification index of the maximal abelian subextension of the field K Ka/Qp.

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.


2018 ◽  
Vol 33 (2) ◽  
pp. 307
Author(s):  
Owais Ahmad ◽  
Neyaz Ahmad Sheikh

The main objective of this paper is to provide complete characterization of multigenerator Gabor frames on a periodic set $\Omega$ in $K$. In particular, we provide some necessary and sufficient conditions for the multigenerator Gabor system to be a frame for $L^2(\Omega)$. Furthermore, we establish the complete characterizations of multigenerator Parseval Gabor frames.


1970 ◽  
Vol 22 (2) ◽  
pp. 297-307 ◽  
Author(s):  
Melvin Band

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.


2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Ján Maňuch

International audience Let X be a two-element set of words over a finite alphabet. If a bi-infinite word possesses two X-factorizations which are not shiftequivalent, then the primitive roots of the words in X are conjugates. Note, that this is a strict sharpening of a defect theorem for bi-infinite words stated in \emphKMP. Moreover, we prove that there is at most one bi-infinite word possessing two different X-factorizations and give a necessary and sufficient conditions on X for the existence of such a word. Finally, we prove that the family of sets X for which such a word exists is parameterizable.


2018 ◽  
Vol 11 (1) ◽  
pp. 35
Author(s):  
Pairote Yiarayong ◽  
Manoj Siripitukdet

Let $\phi: S(M) \rightarrow S(M) \cup \left\lbrace \emptyset\right\rbrace $ be a function where $S(M)$ is the set of all submodules of $M$. In this paper, we extend the concept of $\phi$-$2$-absorbing primary submodules to the context of $\phi$-$2$-absorbing semi-primary submodules. A proper submodule $N$ of $M$ is called a $\phi$-$2$-absorbing semi-primary submodule, if for each $m \in M$ and $a_{1}, a_{2}\in R$ with $a_{1}a_{2}m \in N - \phi(N)$, then $a_{1}a_{2}\in \sqrt{(N : M)}$ or  $a_{1}m \in N$ or $a^{n}_{2}m\in N$, for some positive integer $n$. Those are extended from $2$-absorbing primary, weakly $2$-absorbing primary, almost $2$-absorbing primary, $\phi_{n}$-$2$-absorbing primary, $\omega$-$2$-absorbing primary and $\phi$-$2$-absorbing primary submodules, respectively. Some characterizations of $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-$2$-absorbing semi-primary submodules are obtained. Moreover, we investigate relationships between $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-primary submodules of modules over commutative rings. Finally, we obtain necessary and sufficient conditions of a $\phi$-$\phi$-$2$-absorbing semi-primary in order to be a $\phi$-$2$-absorbing semi-primary.


Author(s):  
G.A. Pinto

An ordered regular semigroup, , is said to be principally ordered if for every  there exists . A principally ordered regular semigroup is pointed if for every element,  we have . Here we investigate those principally ordered regular semigroups that are eventually pointed in the sense that for all  there exists a positive integer, , such that . Necessary and sufficient conditions for an eventually pointed principally ordered regular semigroup to be naturally ordered and to be completely simple are obtained. We describe the subalgebra of  generated by a pair of comparable idempotents  and such that . 


1970 ◽  
Vol 54 (388) ◽  
pp. 113-115
Author(s):  
R. L. Goodstein

We consider the problem of finding necessary and sufficient conditions for a positive integer to be the sum of an arithmetic progression of positive integers with a given common difference, starting with the case when the common difference is unity.


1975 ◽  
Vol 18 (1) ◽  
pp. 155-156 ◽  
Author(s):  
William A. Webb

Problems involving Egyptian fractions (rationals whose numerator is 1 and whose denominator is a positive integer) have been extensively studied. (See [1] for a more complete bibliography). Some of the most interesting questions, many still unsolved, concern the solvability ofwhere k is fixed.In [2] Rav proved necessary and sufficient conditions for the solvabilty of the above equation, as a consequence of some other theorems which are rather complicated in their proofs. In this note we give a short, elementary proof of this theorem, and at the same time generalize it slightly.


2010 ◽  
Vol 75 (8) ◽  
pp. 1093-1098 ◽  
Author(s):  
Damir Vukicevic ◽  
Jelena Djurdjevic ◽  
Ivan Gutman

The Kekul? structure count K of fluoranthene congeners is studied. It is shown that for such polycyclic conjugated ?-electron systems, either K = 0 or K ? 3. Moreover, for every t ? 3, there are infinitely many fluoranthene congeners having exactly t Kekul? structures. Three classes of Kekul?an fluoranthenes are distinguished: (i) ?0 - fluoranthene congeners in which neither the male nor the female benzenoid fragment has Kekul? structures, (ii) ?m - fluoranthene congeners in which the male benzenoid fragment has Kekul? structures, but the female does not, and (iii) ?fm - fluoranthene congeners in which both the male and female benzenoid fragments have Kekul? structures. Necessary and sufficient conditions are established for each class, ? = ?0, ?m, ?fm, such that for a given positive integer t, there exist fluoranthene congeners in ? with the property K = t.


1986 ◽  
Vol 9 (4) ◽  
pp. 801-806 ◽  
Author(s):  
R. A. Mollin ◽  
P. G. Walsh

A powerful number is a positive integernsatisfying the property thatp2dividesnwhenever the primepdividesn; i.e., in the canonical prime decomposition ofn, no prime appears with exponent 1. In [1], S.W. Golomb introduced and studied such numbers. In particular, he asked whether(25,27)is the only pair of consecutive odd powerful numbers. This question was settled in [2] by W.A. Sentance who gave necessary and sufficient conditions for the existence of such pairs. The first result of this paper is to provide a generalization of Sentance's result by giving necessary and sufficient conditions for the existence of pairs of powerful numbers spaced evenly apart. This result leads us naturally to consider integers which are representable as a proper difference of two powerful numbers, i.e.n=p1−p2wherep1andp2are powerful numbers with g.c.d.(p1,p2)=1. Golomb (op.cit.) conjectured that6is not a proper difference of two powerful numbers, and that there are infinitely many numbers which cannot be represented as a proper difference of two powerful numbers. The antithesis of this conjecture was proved by W.L. McDaniel [3] who verified that every non-zero integer is in fact a proper difference of two powerful numbers in infinitely many ways. McDaniel's proof is essentially an existence proof. The second result of this paper is a simpler proof of McDaniel's result as well as an effective algorithm (in the proof) for explicitly determining infinitely many such representations. However, in both our proof and McDaniel's proof one of the powerful numbers is almost always a perfect square (namely one is always a perfect square whenn≢2(mod4)). We provide in §2 a proof that all even integers are representable in infinitely many ways as a proper nonsquare difference; i.e., proper difference of two powerful numbers neither of which is a perfect square. This, in conjunction with the odd case in [4], shows that every integer is representable in infinitely many ways as a proper nonsquare difference. Moreover, in §2 we present some miscellaneous results and conclude with a discussion of some open questions.


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