In this article, we consider compositions of positive integers with 2
and 3
. We see that these compositions lead us to results that involve Padovan numbers, and we give some tiling models of these ...compositions. Moreover, we examine some tiling models of the compositions related to the Padovan polynomials and prove some identities using the tiling model’s method. Next, we obtain various identities of the compositions of positive integers with 2
and 3
related to the Padovan numbers. The number of palindromic compositions of this type is determined, and some numerical arithmetic functions are defined. Finally, we provide a table that compares all of the results obtained from compositions of positive integers with 2
and 3
The polynomial expansion method is a useful tool for solving both the direct and inverse Stokes problems, which together with the pointwise collocation technique is easy to derive the algebraic ...equations for satisfying the Stokes differential equations and the specified boundary conditions. In this paper we propose two novel numerical algorithms, based on a third–first order system and a third–third order system, to solve the direct and the inverse Cauchy problems in Stokes flows by developing a multiple-scale Pascal polynomial method, of which the scales are determined a priori by the collocation points. To assess the performance through numerical experiments, we find that the multiple-scale Pascal polynomial expansion method (MSPEM) is accurate and stable against large noise.
•We solve direct and inverse Cauchy–Stokes problems.•We propose a third–first order system and a third–third order system.•A multiple-scale Pascal polynomial method is developed.•The scales are determined by the collocation points.•The multiple-scale Pascal polynomial expansion method is accurate and stable against large noise.
We give a formula for
f
(
η
)
,
where
f
:
C
→
C
is a continuously differentiable function satisfying
f
(
z
¯
)
=
f
(
z
)
¯
,
and
η
is a dual quaternion. Note this formula is straightforward or well ...known if
η
is merely a dual number or a quaternion. If one is willing to prove the result only when
f
is a polynomial, then the methods of this paper are elementary.
The polynomial expansion method is a useful tool to solve partial differential equations (PDEs). However, the researchers seldom use it as a major medium to solve PDEs due to its highly ...ill-conditioned behavior. We propose a single-scale and a multiple-scale Pascal triangle formulations to solve the linear elliptic PDEs in a simply connected domain equipped with complex boundary shape. For the former method a constant parameter R0 is required, while in the latter one all introduced scales are automatically determined by the collocation points. Then we use the multiple-scale method to solve the inverse Cauchy problems, which is very accurate and very stable against large noise to 20%. Numerical results confirm the validity of the present multiple-scale Pascal polynomial expansion method.
This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of ...words and the sequence (S(n))n≥0 counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for (S(n))n≥0. This leads to a connection with the 2-regular Stern–Brocot sequence and the sequence of denominators occurring in the Farey tree. Then we extend our construction to the Zeckendorf numeration system based on the Fibonacci sequence. Again our tree structure permits us to obtain recurrence relations for and the F-regularity of the corresponding sequence.
In Autonomous Mobile Networks (AMNs), nodes regularly collaborate as well as transmit all packets to facilitate out of range transaction. Though, in unfriendly surroundings, several nodes might ...reject for preserve their own senders or for intentionally disrupting regular communications. Ina addition, the node mobility to handles the active and passive attacks. To overcome these problems, in this article Detect and Prevent Active and Passive Attack in Autonomous Mobile Network (DPAP) is introduced. Here, faithful forwarder node is chosen based on the vicinity information as well as connectivity of node. The active attacker node is detected by three hop verification method. In addition, the Binomial Pascal triangle function (BPTF) is applied for concealing the node uniqueness as a result; it recognizes the passive attack in the AMN. The DPAP approach is measured by applying a network simulator-2 for recognizing the active and passive attacker in the AMN. Simulation output illustrates the DPAP offer better throughput and detection ratio in the AMN.
Moessner's theorem describes a procedure for generating a sequence of n integer sequences that lead unexpectedly to the sequence of nth powers 1n, 2n, 3n, … . Several generalizations of Moessner's ...theorem exist. Recently, Kozen and Silva gave an algebraic proof of a general theorem that subsumes Moessner's original theorem and its known generalizations. In this note, we describe the formalization of this theorem that the first author did in Nuprl. On the one hand, the formalization remains remarkably close to the original proof. On the other hand, it leads to new insights in the proof, pointing to small gaps and ambiguities that would never raise any objections in pen and pencil proofs, but which must be resolved in machine formalization.
We show that the $n$'th diagonal sum of Barry's modified Pascal triangle can be described as the continuant of the run lengths of the binary representation of $n$. We also obtain an explicit ...description for the row sums.
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