Lagrange's Theorem Research Articles

Speed Enhancement Technique for Pulsed Laser Rangefinders Based on Lagrange's Theorem Using an Undersampling Method

A new speed enhancement technique for pulsed laser rangefinders based on Lagrange's theorem in group theory using an undersampling method has been developed. In the undersampling method, frequency conversion for high-resolution ranging and digitizing are conducted by sampling a reference frequency signal at the timings of the reception of pulsed light from the target. In the present work, the rangefinder generates different sampling intervals of the reference frequency signal: different numbers of sampling points within the period of a reference signal, over a wide range. This is accomplished by slightly changing the period of the pulsed light emitted, without changing the synthesizer frequency which generates the period. This technique requires a minimum of additional hardware. In this paper, we describe the detail of the selection of the number of sampling points based on Lagrange's theorem. And we demonstrate a possibility of expanding the sampling interval to the point where an aliasing of the harmonic components of the reference signal occurs by simulations that focus on the calculation of the phase of the fundamental frequency of the reference signal. And we report on the results of rangefinder experiments for a reduction in the number of the sampling points. We have achieved a 10-fold enhancement of speed by selecting 10 sampling points over the results from the previous studies that had 100 sampling points within a period of a reference signal. And we have confirmed that the reduction in sampling points has a very little influence on the linearity, which is an acceptable trade-off for achieving the speed enhancement. This technique, based on Lagrange's theorem in group theory, allows us to control the minimum number of samplings required to calculate distances, so that high-speed data acquisition for coarse measurements and normal-speed data acquisition for fine measurements become selectable. Such a system with high flexibility in measurement modes has been developed.

Lagrange's Theorem for Hopf Monoids in Species

We prove Lagrange's theorem for Hopf monoids in the category of connected species. We deduce necessary conditions for a given subspecies $\textrm{k}$ of a Hopf monoid $\textrm{h}$ to be a Hopf submonoid: each of the generating series of $\textrm{k}$ must divide the corresponding generating series of $\textrm{k}$ in ℕ〚x〛. Among other corollaries we obtain necessary inequalities for a sequence of nonnegative integers to be the sequence of dimensions of a Hopf monoid. In the set-theoretic case the inequalities are linear and demand the non negativity of the binomial transform of the sequence. Nous prouvons le théorème de Lagrange pour les monoïdes de Hopf dans la catégorie des espèces connexes. Nous déduisons des conditions nécessaires pour qu'une sous-espèce $\textrm{k}$ d'un monoïde de Hopf $\textrm{h}$ soit un sous-monoïde de Hopf: chacune des séries génératrices de $\textrm{k}$ doit diviser la série génératrice correspondante de $\textrm{h}$ dans ℕ〚x〛. Parmi d'autres corollaires nous trouvons des inégalités nécessaires pour qu'une suite d'entiers soit la suite des dimensions d'un monoïde de Hopf. Dans le cas ensembliste les inégalités sont linéaires et exigent que la transformée binomiale de la suite soit non négative.

Continued fractions without fractions: Lagrange theorem and Pell equations

Quadratic irrationals and their representation as continued fractions are investigated by means of proper tools of Discrete Dynamical Systems theory. This is done by recasting the process of generation of digits in the continued fraction expansion of a real number as a suitable discrete nonlinear system whose asymptotic behavior is to be studied. Such an approach allows, on the one hand, to retrieve well known results in the literature, the most relevant being the Lagrange theorem, and on the other hand it aims to give more insight into some related issues.

A Curious Proof of Fermat's Little Theorem

Fermat's little theorem states that for p prime and a e Z, p divides ap a. This result is of huge importance in elementary and algebraic number theory. For instance, with its help we obtain the so-called Frobenius automorphism of a finite field ¥pn over ¥p. This theorem has many interesting and sometimes unexpected proofs. One classical proof is based upon properties of binomial coefficients. In fact, (d + ')p dp 1 = EfJi1 {')#. Since (p) = jr^ is divisible by p for 1 < i < p 1, then {d + 1) dp 1 is divisible by p. Summing this over d = 1 , 2, . . . , a 1 , we obtain the desired result. Another classical proof is based upon Lagrange's theorem, which states that the order of an element of a finite group divides the group order. Applying this theorem to the multiplicative group of a finite field ¥p we obtain the result immediately. Several other proofs can be found at [2]. Nevertheless, in all of these proofs one or another analogue of the Euclidean algorithm (hence arithmetic) is being used. In this short note we present a proof which was found as a side result of another, unrelated problem (which is the case, maybe, with many such curious proofs). Surprisingly, arithmetic, group theory, and the properties of binomial coefficients do not manifest at all.

A General Lagrange Theorem

The ordinary continued fractions expansion of a real number is based on the Euclidean division. Variants of the latter yield variants of the former, all encompassed by a more general Dynamical Systems framework. For all these variants the Lagrange Theorem holds: a number has an eventually periodic expansion if and only if it is a quadratic irrational. This fact is surely known for specific expansions, but the only proof for the general case that I could trace in the literature follows as an implicit corollary from much deeper results by Boshernitzan and Carroll on interval exchange transformations. It may then be useful to have at hand a simple and virtually computation-free proof of a general Lagrange Theorem.

A General Lagrange Theorem

(2009). A General Lagrange Theorem. The American Mathematical Monthly: Vol. 116, No. 1, pp. 70-74.

On a multidimensional generalization of Lagrange's theorem on continued fractions

We prove a multidimensional analogue of the classical Lagrange theorem on continued fractions using Klein polyhedra as a multidimensional generalization of continued fractions.

Four proofs of the converse of the Chinese Remainder Theorem

Four proofs, designed for classroom use in varying levels of courses on abstract algebra, are given for the converse of the classical Chinese Remainder Theorem over the integers. In other words, it is proved that if m and n are integers greater than 1 such that the abelian groups and are isomorphic, then m and n are relatively prime. The first proof, which involves the order of an element in a finite group, can be given early in a first course on abstract algebra. The second proof, which can be given later in such a course, depends on Lagrange's Theorem and the Second Isomorphism Theorem. The final two proofs, which involve tensor products and exterior powers, can be given in more advanced courses that develop or assume multilinear algebra. The latter two proofs generalize to give the converse of the Chinese Remainder Theorem for modules over commutative rings.

Some classes of finite groups and mutually permutable products

This paper is devoted to the study of mutually permutable products of finite groups. A factorised group G = A B is said to be a mutually permutable product of its factors A and B when each factor permutes with every subgroup of the other factor. We prove that mutually permutable products of Y -groups (groups satisfying a converse of Lagrange's theorem) and SC-groups (groups whose chief factors are simple) are SC-groups, by means of a local version. Next we show that the product of pairwise mutually permutable Y -groups is supersoluble. Finally, we give a local version of the result stating that when a mutually permutable product of two groups is a PST-group (that is, a group in which every subnormal subgroup permutes with all Sylow subgroups), then both factors are PST-groups.

The Hahn–Banach–Lagrange theorem†

This article is about a new version of the Hahn–Banach theorem, which we will call the “Hahn–Banach–Lagrange theorem”, since it deals very effectively with certain problems of Lagrange type, as well as giving numerous results in functional analysis, convex analysis, and monotone operator theory. We will discuss several of these results in this article. †Dedicated to Diethard Pallaschke, on the occasion of his 65th birthday.

A Simple Recurrence for Covers of the Sphere With Branch Points of Arbitrary Ramification

The problem of counting ramied covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factor- ing a permutation of specied cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity. Goulden and Jackson have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the Join-Cut equation. Recently, Bousquet-M· and Schaeffer have found the number of minimal, transitive factorizations of a permutation into arbitrary unspecied factors. This was proved by a purely combinatorial argument, based on a direct bijection between factorizations and certain objects called m-Eulerian trees. In this paper, we give a simple partial differential equation for Bousquet-M· elou and Schaef- fer's generating series, and for Goulden and Jackson's generating series, as well as a new proof of the result by Bousquet-M· and Schaeffer. We apply algebraic methods based on Lagrange's theorem, and combinatorial methods based on a new use of Bousquet-M· and Schaeffer's m-Eulerian trees.

Closure to “Exact Solutions for Sequent Depths Problem” by Prabhata K. Swamee and Pushpa N. Rathie

The writers are very grateful to the discussers for their discussions. Following are the replies to the comments by Rajesh Srivastava: 1. Although Eqs. 10 and 11 are explicit and valid for the full range of the independent variable fL, these equations are not exact. In addition to introducing the Lagrange theorem to the engineering community, the paper is aimed at obtaining an exact solution of a system of implicit equations. Since the solution is in the form of an infinite series, its accuracy can be varied by truncating the series at an appropriate term. 2. The writers thank the discusser for conducting a detailed error analysis. A perusal of Fig. 1 of the discussion shows that Eq. 10 exhibits a large error for very large fL corresponding to an extremely large prejump Froude number. For such a large prejump Froude number, the flow becomes

Lagrange's theorem for Moufang loops

We prove that the order of any subloop of a finite Moufang loop is a factor of the order of the loop, thus obtaining an analog of Lagrange's theorem for finite Moufang loops.

A C∞ Lagrange Inversion Theorem

Naturally there are questions of existence and convergence associated with (2). We will discuss these in due course. The purpose of this note is to present a short and transparent derivation of (2) based on the Laplace transform under hypotheses less stringent than are usually imposed. Krantz and Parks [6] present a history of Lagrange's inversion theorem. Several different derivations of (2) can be found in the modem literature:

Espaces d'écoulements dits « universels », 3

Universal motions with uniform steady vorticity form a corolla of linear spaces derived from rigid body motions. Closely related to potential flows, they satisfy two extensions of Lagrange theorem, are investigated with the help of complex functions, as stand celebrated when be plane. They take place in hydrodynamics, aerodynamics, geophysics, astrophysics, turbulence, physics of plasmas and superfluid helium. In all the cases, arbitrary unsteady span-wise translations permit to generalise as well as to exhibit helical or 3D universal motions. Three misunderstood periodic flows illustrate our purpose, as they approach shear instabilities in numerous fluids. To cite this article: M. Bouthier, C. R. Mecanique 332 (2004).

The quadratic Waring–Goldbach problem

It is conjectured that Lagrange's theorem of four squares is true for prime variables, i.e. all positive integers n with n≡4 ( mod 24) are the sum of four squares of primes. In this paper, the size for the exceptional set in the above conjecture is reduced to O(N 3 8 +ε ) .

THE COMMUTATOR SUBGROUP AND CLT(NCLT) GROUPS

The commutator subgroup G' can indicate if a finite group G is a CUT (Converse Lagrange's Theorem) group or an NCLT (Non-Converse Lagrange's Theorem) group. We give general results and some examples of their application to groups of small order.

The Commutator Subgroup and &lt;i&gt;CLT(NCLT)&lt;/i&gt; Groups

The commutator subgroup G' can indicate if a finite group G is a CLT (Converse Lagrange's Theorem) group or an NCLT (Non-Converse Lagrange's Theorem) group. We give general results and some examples of their application to groups of small order.

On Lagrange's Theorem with Prime Variables

On Lagrange's Theorem with Prime Variables

On Lagrange's Theorem with Prime Variables

On Lagrange's Theorem with Prime Variables