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The objective of navigation is to determine the instantaneous location of a moving vehicle. Historically, sailors estimated the speed of a ship by dropping a log into the water and observing the time for it to float from the bow to the stern [1, pp. 21-38]. Combining this speed information with directional data from a compass yields a velocity estimate. Finally, integrating the velocity estimate with respect to time and using the known initial position yields a position estimate at each instant of time. This procedure is called dead reckoning [2, p. 30], [3] or, when performed by animals without the benefit of calculus and digital computation, path integration [4]. Although not all navigation is based on dead reckoning-for example, celestial navigation uses the stars for position fixing-it is widely used due to its ability to utilize speed, direction, and time data from a variety of sensors. By computing position from velocity data, dead reckoning is an integration process. It is important to stress, however, that, because of the lack of observability, dead reckoning cannot take advantage of an observer and, thus, cannot stabilize the integrator dynamics. Therefore, dead reckoning is an inherently unstable process, regardless of how the velocity and direction data are acquired.

In the paper, employing methods and techniques in analysis and linear algebra, the authors find a simple formula for computing an interesting Hessenberg determinant whose elements are products of binomial coefficients and falling factorials, derive explicit formulas for computing some special Hessenberg and tridiagonal determinants, and alternatively and simply recover some known results.

Although controllability and observability are distinct properties, one of the fundamental-and most attractive-results of our field is the fact that (A, B) is controllable if and only if f (AT,BT) is observable. This duality provides a deep linkage between the linear-quadratic regulator (LQR), which seeks a feedback gain K such that A + BK is asymptotically stable, and the linear-quadratic estimator (LQE), which seeks an output-error-injection gain F such that A+FC is asymptotically stable. In the case of LQR, the controllability of (A, B) implies that there exists a feedback gain K that arbitrarily places the eigenvalues of A+BK, thus facilitating closed-loop asymptotic stability. In the dual case of LQE, the observability of (A, C) implies that there exists an error-injection gain F that arbitrarily places the eigenvalues of A + FC, thus facilitating closedloop asymptotic stability of the error dynamics. A key distinction worth noting is that A + BK is the dynamics matrix of a physical feedback loop, whereas A + FC is the dynamics matrix of a nonphysical error system.

One of the many things that humans and many animals are very good at is picking up objects. Teaching robots to pick up objects is not so easy, however, for at least two reasons. First, the coefficient of friction may be unknown, and therefore, an object may slip through the robot's grasp. Although this uncertainty can be overcome by using a tighter grip, another issue arises, namely, that excessive force may crush the object. Furthermore, the length of the object must also be accounted for since the ultimate goal is to achieve a slip distance that is less than the length of the object. Consequently, reliable gripping is a control problem that requires a careful tradeoff between the possibility of dropping an object and the potential for crushing it.

We show that the formula of Faa di Bruno for the nth derivative of a composite function can be applied to deduce various combinatorial identities involving Lah numbers, Stirling numbers and harmonic numbers. One of our results states that for all integers m, n with $$m\ge n\ge 1$$ , we have $$\begin{aligned}&2^n {m\atopwithdelims ()n} (2 H_{2m-n}-H_{m-n})\\&\quad =\sum _{k=0}^{\left\lfloor n/2 \right\rfloor } (-1)^k {m\atopwithdelims ()k} {2m-2k\atopwithdelims ()n-2k} (2H_{2(m-k)}-H_{m-k}), \end{aligned}$$ where $$H_k=1+1/2+\cdots + 1/k$$ denotes the kth harmonic number. Moreover, we present connections between absolutely and completely monotonic functions and we provide a three-parameter class of completely monotonic functions.

This paper analyzes the dynamics of a bead sliding along a rotating horizontal wire whose rate of rotation varies due to the motion of the bead. Legendre’s elliptic integrals are used to obtain an analytical solution of the equation of motion given by a nonlinear second-order differential equation. The analytical solution is used to determine the terminal velocity of the bead and the asymptotic angle of the wire.

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy’s integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

Consider a sequence $$(a_n)_{n\ge 1}$$ of complex numbers such that for some positive number p we have $$\lim _{n\rightarrow \infty }\frac{1}{n^p}\left( \sum _{k=1}^na_k\right) =\ell \in \mathbb {C}$$ . We prove that, under some mild conditions on the sequence (positivity or absolute boundedness in the mean, etc.), we have $$\begin{aligned} \lim _{n\rightarrow \infty } \frac{1}{n^{q}}\sum _{k=1}^nk^{q-p}a_kf\left( \frac{k}{n}\right) = p\ell \int _0^1x^{q-1}f(x)dx \end{aligned}$$ for all $$q>0$$ and all Riemann integrable functions $$f:[0,1]\rightarrow \mathbb {C}$$ . Some applications to probability theory and to number theory are also discussed.

A necklace or bracelet is \textit{colorful} if no pair of adjacent beads are the same color. In addition, two necklaces are \textit{equivalent} if one results from the other by permuting its colors, and two bracelets are \textit{equivalent} if one results from the other by either permuting its colors or reversing the order of the beads; a bracelet is thus a necklace that can be turned over. This note counts the number $K(n)$ of non-equivalent colorful necklaces and the number $K'(n)$ of colorful bracelets formed with $n$-beads in at most three colors. Expressions obtained for $K'(n)$ simplify expressions given by OEIS sequence A114438, while the expressions given for $K(n)$ appear to be new and are not included in OEIS., Comment: 19 pages, 2 figures

In this note, we recall Kummer's Fourier series expansion of the 1-periodic function that coincides with the logarithm of the Gamma function on the unit interval $(0,1)$, and we use it to find closed forms for some numerical series related to the generalized Stieltjes constants, and some integrals involving the function $x\mapsto \ln \ln(1/x)$.

When Newton's method, or Halley's method is used to approximate the $p${th} root of $1-z$, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010)., 10 pages

The geometric factor for the resistivity measurement of a thin disk, given in a previous work as a slowly converging series, is reworked and reduced to a simple form. On the other hand, the correction, when contacts are assumed of vanishing width, is also calculated analytically and discussed. It is found not to exceed 1.5% when λ , the ratio of distances between potential contacts and current contacts, is equal to 1. Our new expression is applied to determine the room temperature resistivity of two metallic thin disks.

The expression of the transverse resistance for an ohmic parallelepipedic layered conductor, measured with strip contacts, extended along its width, is a slowly convergent series. This series is reworked and transformed to the sum of an analytical part and an exponentially convergent series, which reduces considerably the number of terms needed for the numerical evaluation. In addition, an asymptotic formula is obtained, valid for Γ Γ is the effective anisotropy. This formula is used to determine the room temperature resistivity anisotropy of two small Γ layered superconductors, which are 2H–NbSe 2 and 2H–TaSe 2 .

We prove an inequality on positive real numbers, that looks like a reverse to the well-known Hilbert inequality, and we use some unusual techniques from Fourier analysis to prove that this inequality is optimal., Comment: 7 pages

In this note, a general formula is proved. It expresses the integral on the line of the product of a function $f$ and a periodic function $g$ in terms of the Fourier transform of $f$ and the Fourier coefficients of $g$. This allows the evaluation of some oscillatory integrals.

In this note a general result is proved that can be used to evaluate exactly a class of highly oscillatory integrals., Comment: 12 pages, 1 figure

In this note we prove optimal inequalities for bounded functions in terms of their deviation from their mean. These results extend and generalize some known inequalities due to Thong (2011) and Perfetti (2011), 7 pages

Using methods from classical analysis, sharp bounds for the ratio of differences of Power Means are obtained. Our results generalize and extend previous ones due to S. Wu(2005), and to S. Wu and L. Debnath., 1 figure

We prove a general result on the factorization of matrix-valued analytic functions. We deduce that if $(E_0,E_1)$ and $(F_0,F_1)$ are interpolation pairs with dense intersections, then under some conditions on the spaces $E_0$, $E_1$, $F_0$ and $F_1$, we have $$ [E_0\hat\otimes F_0,E_1\hat\otimes F_1]_t= [E_0, E_1]_t\hat\otimes[F_0, F_1]_t, 0 < t< 1.$$ We find also conditions on the spaces $E_0$, $E_1$, $F_0$ and $F_1$, so that the following holds $$ [E_0\wcheck\otimes F_0,E_1\wcheck\otimes F_1]_t= [E_0,E_1]_t\wcheck\otimes [F_0,F_1]_t, 0, Comment: 26 pages

It is shown in this work that the electric resistance expression for an anisotropic circular disk diverges if a finite dimension is assumed for a current contact and if the position of this contact approaches the axis of the least resistivity. Accordingly, the optimal position of the current contacts is along the axis of the highest resistivity. This expression is mainly used to determine the resistivity anisotropy; a new contacts configuration for the resistivity anisotropy determination is suggested.

11056 [2004, 64]. Proposed by John H. Lindsey II, Cambridge, MA. Let n be a positive integer, and let p be a polynomial of degree at most n. Show that on any nonempty finite real interval I, maxxe/ Ip(x)\ is at most 4n2 times the average on I of Ip(x)|. Solution by Omran Kouba, Higher Institute for Applied Sciences and Technology, Damascus, Syria. We establish a stronger bound: maxxE/ Ip(x)l is at most (n + 1)2 times the average on I of Ip(x) . By an affine change of variable, we may assume that I = [-1, 1]. Recall that the Legendre polynomials, defined for n E N by

In this note we present a proof of the combinatorial nullstellensatz using simple arguments from linear algebra.