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Bell System Technical Journal, 51: 2. February 1972 pp 507-542. Controllability and Observability in Linear Time-Variable Networks With Arbitrary Symmetry Groups. (Rubin, H.; Meadows, H.E.)

Let G be the (special) affine group, semidirect product of SL_n and C^n. In this paper we study the representation theory of G and in particular the question of rationality for V/G where V is a generically free G-representation. We show that the answer to this question is positive if the dimension of V is sufficiently large and V is indecomposable. We have a more precise theorem if V is a two-step extension 0 -> S -> V -> Q -> 0 with S, Q completely reducible.

We show that if $T$ is an isometry (as metric spaces) from an open subgroup of the invertible group $A^{-1}$ of a unital Banach algebra $A$ onto an open subgroup of the invertible group $B^{-1}$ of a unital Banach algebra $B$, then $T$ is extended to a real-linear isometry up to translation between these Banach algebras. We consider multiplicativity or unti-multiplicativity of the isometry. Note that a unital linear isometry between unital semisimple commutative Banach algebra need be multiplicative. On the other hand, we show that if $A$ is commutative and $A$ or $B$ are semisimple, then $(T(e_A))^{-1}T$ is extended to a isometrical real algebra isomorphism from $A$ onto $B$. In particular, $A^{-1}$ is isometric as a metric space to $B^{-1}$ if and only if they are isometrically isomorphic to each other as metrizable groups if and only if $A$ is isometrically isomorphic to $B$ as a real Banach algebra; it is compared by the example of \.Zelazko concerning on non-isomorphic Banach algebras with homeomorphically isomorphic invertible groups. Maps between standard operator algebras are also investigated.

Let $V$ be a faithful $G$-module for a finite group $G$ and let $p$ be a prime dividing $|G|$. An orbit $v^G$ for the action of $G$ on $V$ is $p$-regular if $|v^G|_p=|G:\bC_G(v)|_p=|G|_p$. Zhang asks the following question in \cite{Zhang}. Assume that a finite solvable group $G$ acts faithfully and irreducibly on a vector space $V$ over a finite field $\FF$. If $G$ has a $p$-regular orbit for every prime $p$ dividing $|G|$, is it true that $G$ will have a regular orbit on $V$? In \cite{LuCao}, L\"{u} and Cao construct an example showing that the answer to this question is no, however the example itself is not correct. In this paper, we study Zhang's question in detail. We construct examples showing that the answer to this question is no in general. We also prove the following result. Assume a finite solvable group $G$ of odd order acts faithfully and irreducibly on a vector space $V$ over a field of odd characteristic. If $G$ has a $p$-regular orbit for every prime $p$ dividing $|G|$, then $G$ will have a regular orbit on $V$.

We consider the modified (or twisted) Yang-Baxter equations for the $SL_{q}(N)$ groups and $SL_{q}(N|M)$ supergroups. The general solutions for these equations are presented in the case of the linear quantum (super)groups. The introduction of spectral parameters in the twisted Yang-Baxter equation and its solutions are also discussed.

We reconstruct derived Witt groups via special linear algebraic cobordism. There is a morphism of ring cohomology theories which sends the canonical Thom class in special linear cobordism to the Thom class in the derived Witt groups. We show that for every smooth variety X this morphism induces an isomorphism between MSL^{*,*}(X)[h^{-1}] with the "extended" coefficient ring MSL^{4*,2*}(pt) -> W^{2*}(pt) and Laurent polynomial ring over the derived Witt groups W^*(X), where h is the stable Hopf map. This result is an analogue of the result by Panin and Walter reconstructing hermitian K-theory using symplectic algebraic cobordism.

Motivated by Bownik and Speegle's result on linear independence of wavelet Parseval frames, we consider affine systems (analogous to wavelet systems) defined on a second countable, locally compact abelian group $G$, where the translations are replaced by the action of a countable, discrete subgroup $\Gamma$ of $ G$ acting as a group of unitary operators on $L^2(G)$. The dilation operation in the wavelet setting is replaced by integer powers of a unitary operator $\delta$ onto $L^2(G)$. We show that, under some compatibility conditions between $\delta$ and the action of the group $\Gamma$, the linear independence of the translates of any function in $L^2(G)$ by elements of $\Gamma$ implies the linear independence of affine Parseval frames in $L^2(G)$.

"Irreducible Linear Homogeneous Groups Whose Orders Are Powers of a Prime" is an article from Transactions of the American Mathematical Society, Volume 8 . View more articles from Transactions of the American Mathematical Society . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1986237

"Determination of the Finite Quaternary Linear Groups" is an article from Transactions of the American Mathematical Society, Volume 14 . View more articles from Transactions of the American Mathematical Society . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1988774

"Group-Characters of Various Types of Linear Groups" is an article from American Journal of Mathematics, Volume 29 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370015

"Representation of Linear Groups as Transitive Substitution Groups" is an article from American Journal of Mathematics, Volume 23 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370134

"Representation of Linear Groups as Transitive Substitution Groups" is an article from American Journal of Mathematics, Volume 23 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370134

We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of non positive curvature and which do not hold for arbitrary characteristic zero linear groups. In particular if such a linear group is finitely generated then centralisers virtually split and all finitely generated abelian subgroups are undistorted. If further the group is virtually torsion free (which always holds in characteristic zero) then we have a strong property on small subgroups: any subgroup either contains a non abelian free group or is finitely generated and virtually abelian, hence also undistorted. We present applications, including that the mapping class group of a surface having genus at least 3 has no faithful linear representation which is complex unitary or over any field of positive characteristic.

This study investigates the population sensitivity of the commonly used linear equating methods in the Non-Equivalent-groups with an Anchor Test (NEAT) design: the Tucker, the Levine observed-score, and the chain linear methods. For a detailed analysis of the subject, we apply three distinctive approaches to a real data set from a NEAT design: a) the RMSD index for the NEAT design of von Davier, Holland, and Thayer (2004); b) the parallel-linking system of Dorans and Holland (2000), and c) the pseudo-NEAT design approach of von Davier, Holland, and Thayer (2003). The data set is used to illustrate the derivations on male and female subpopulations. Comparisons of the results obtained using linear and equipercentile equating are also presented. For this particular data set the Levine function seems to vary less across subpopulations than the other methods; the Tucker method seems to vary the most.

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"On the Order of Linear Homogeneous Groups (Second Paper)" is an article from Transactions of the American Mathematical Society, Volume 5 . View more articles from Transactions of the American Mathematical Society . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1986460

Journal of Research of the National Bureau of Standards

Let $G$ be a reductive group over an algebraically closed field $k$, $H$ a subgroup normalised by a maximal torus, and $X$ an affine $k$-variety acted on by $G$. The Popov-Pommerening conjecture from 1985 says that the invariant ring $k[X]^H$ is finitely generated. We prove the conjecture for linear algebraic groups $G$ over $k=\CC$ and give a geometric description of a generating set of $\CC[X]^H$.

"A Theorem Concerning the Invariants of Linear Homogeneous Groups, with some Applications to Substitution-Groups" is an article from Transactions of the American Mathematical Society, Volume 5 . View more articles from Transactions of the American Mathematical Society . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1986275

"A Theorem Concerning the Invariants of Linear Homogeneous Groups, with some Applications to Substitution-Groups" is an article from Transactions of the American Mathematical Society, Volume 5 . View more articles from Transactions of the American Mathematical Society . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1986275

We prove that the irreducible decomposition of the permutation representation of GL(n,q) on GL(n,q)/GL(n-m,q) stabilizes for large n. We deduce, as a consequence, a representation stability theorem for finitely generated VIC-modules.

Kontsevich and Soibelman defined the Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety can produce an example of such a category, whose corresponding Donaldson-Thomas invariants are encoded by a special formal automorphism of the cluster variety, known as the Donaldson-Thomas transformation. In this paper we prove a conjecture of Goncharov and Shen in the case of $\mathrm{GL}_n$, which describes the Donaldson-Thomas transformation of the double quotient of the double Bruhat cells $H \backslash \mathrm{GL}_n^{u,v}/H$ where $H$ is a maximal torus, as a certain explicit cluster transformation related to Fomin-Zelevinsky's twist map. Our result, combined with the work of Gross, Hacking, Keel, and Kontsevich, proves the duality conjecture of Fock and Goncharov in the case of $H\backslash \mathrm{GL}_n^{u,v}/H$.

In this paper we study a classification of linear systems on Lie groups with respect to the conjugacy of the corresponding flows. We also describe stability according to Lyapunov exponents.

We revisit an order-six linear differential operator having a solution which is a diagonal of a rational function of three variables. Its exterior square has a rational solution, indicating that it has a selected differential Galois group, and is actually homomorphic to its adjoint. We obtain the two corresponding intertwiners giving this homomorphism to the adjoint. We show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition, already underlined in a previous paper, in terms of order-two self-adjoint operators. From these results, we deduce a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We then generalize the previous decomposition to decompositions in terms of an arbitrary number of self-adjoint operators of the same parity order. This yields an infinite family of linear differential operators homomorphic to their adjoint, and, thus, with a selected differential Galois group. We show that the equivalence of such operators is compatible with these canonical decompositions. The rational solutions of the symmetric, or exterior, squares of these selected operators are, noticeably, seen to depend only on the rightmost self-adjoint operator in the decomposition. These results, and tools, are applied on operators of large orders. For instance, it is seen that a large set of (quite massive) operators, associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained recently by P. Lairez, correspond to a particular form of the decomposition detailed in this paper.

We prove that the residual girth of any finitely generated linear group is at most exponential. This means that the smallest finite quotient in which the $n$-ball injects has at most exponential size. If the group is also not virtually nilpotent, it follows that the residual girth is precisely exponential.

Let $q$ be a prime power, $G=GL_n(q)$ and let $U\leqslant G$ be the subgroup of (lower) unitriangular matrices in $G$. For a partition $\lambda$ of $n$ denote the corresponding unipotent Specht module over the complex field $\C$ for $G$ by $S^\lambda$. It is conjectured that for $c\in \Z_{\geqslant 0}$ the number of irreducible constituents of dimension $q^c$ of the restriction $\RRes^{G}_U(S^\lambda)$ of $S^\lambda$ to $U$ is a polynomial in $q$ with integer coefficients depending only on $c$ and $\lambda$, not on $q$. In the special case of the partition $\lambda=(1^n)$ this implies a longstanding (still open) conjecture of Higman \cite{higman}, stating that the number of conjugacy classes of $U$ should be a polynomial in $q$ with integer coefficients depending only on $n$ not on $q$. In this paper we prove the conjecture in the case that $\lambda=(n-m,m)$ $(0\leqslant m \leqslant n/2)$ is a 2-part partition. As a consequence, we obtain a new representation theoretic construction of the standard basis of $S^\lambda$ (over fields of characteristic coprime to $q$) defined by M. Brandt, R. Dipper, G. James and S. Lyle in \cite{brandt2}, \cite{dj1} and an explanation of the rank polynomials appearing there.

We prove that the groups PSL_n(q) are (2,3)-generated for n=9,10 or 11 and all q. Actually, we find out explicit generators x_n and y_n of respective orders 2 and 3, for the groups SL_n(q).

Test data from 200 examinees from the Spanish Teacher Certification Examination and 75 examinees from the French Teacher Certification Examination were used in a study of scale drift in sequentially equated test forms. Using sampling with replacement, 1,000 samples of 100 examinees each for Spanish and 1,000 samples of 50 each for French were created, and each sample was equated using Form A as the base form and equating through form B and form C back to form A. The Spanish forms were then broken down by item into three validated domains while the French forms were categorized into two domains. The forms were then equated by domain and the resulting equated domain scores summed to provide an estimate of total test performance. It was found that the whole test equating provided a better estimate based on scale drift than the summed scores of the equated domains. Nine tables. (Contains six references.) (Author)

In this paper, we prove Lusztig's conjecture for finite special linear groups, i.e., we show that characteristic functions of character sheaves coincide with almost characters up to scalar constants, under the condition that the characteristic of the base field is not too small. We determine those scalars explicitly. Our result gives a method of computing irreducible characters of these groups.

One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps in the standard development, including the self-normalizing property of Borel subgroups.

"Determination of all Groups of Binary Linear Substitutions with Integral Coefficients Taken Modulo 3 and of Determinant Unity" is an article from The Annals of Mathematics, Volume 5 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2007231

"Generalizations in the Theory of Numbers and Theory of Linear Groups" is an article from The Annals of Mathematics, Volume 13 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1968068

Integer linear programs arise in many situations, and solving such problems can be computationally demanding. One way to solve them more e ciently is by exploiting the symmetry within their formulation. This paper proves that the symmetry group for the linear programming relaxation of 2-level orthogonal array problems of strength 2 is a particular semidirect product.

Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G X) < dim G for some definable X subset of G then X contains a torsion point of G. Along the way we develop a general theory for so-called G-linear sets, and investigate definable sets which contain abstract subgroups of G.

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel's equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In the present paper, we give the first known algorithm that calculates the differential Galois group of a third order parameterized linear differential equation.

A $Z_2Z_4$-linear Hadamard code of length $\alpha+2\beta=2^t$ is a binary Hadamard code which is the Gray map image of a $Z_2Z_4$-additive code with $\alpha$ binary coordinates and $\beta$ quaternary coordinates. It is known that there are exactly $\lfloor (t-1)/2 \rfloor$ and $\lfloor t/2 \rfloor$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$, with $\alpha=0$ and $\alpha \not= 0$, respectively, for all $t\geq 3$. In this paper, it is shown that each $Z_2Z_4$-linear Hadamard code with $\alpha=0$ is equivalent to a $Z_2Z_4$-linear Hadamard code with $\alpha \not= 0$; so there are only $\lfloor t/2 \rfloor$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$. Moreover, the order of the monomial automorphism group for the $Z_2Z_4$-additive Hadamard codes and the permutation automorphism group of the corresponding $Z_2Z_4$-linear Hadamard codes are given.

In this paper we give a simple (local) proof of two principal results about irreducible tempered representations of general linear groups over a non-archimedean local division algebra. We give a proof of the parameterization of the irreducible square integrable representations of these groups by segments of cuspidal representations, and a proof of the irreducibility of the tempered parabolic induction. Our proofs are based on Jacquet modules (and the Geometric Lemma, incorporated in the structure of a Hopf algebra). We use only some very basic general facts of the representation theory of reductive p-adic groups (the theory that we use was completed more then three decades ago, mainly in 1970-es). Of the specific results for general linear groups over A, basically we use only a very old result of G.I. Olshanskii, which says that there exist complementary series starting from $Ind(\rho\otimes\rho)$ whenever $\rho$ is a unitary irreducible cuspidal representation. In appendix of the paper "On parabolic induction on inner forms of the general linear group over a non-archimedean local field" of E. Lapid and A. Minguez, there is also a simple local proof of these results, based on a slightly different approach.

In this paper we give a simple (local) proof of two principal results about irreducible tempered representations of general linear groups over a non-archimedean local division algebra. We give a proof of the parameterization of the irreducible square integrable representations of these groups by segments of cuspidal representations, and a proof of the irreducibility of the tempered parabolic induction. Our proofs are based on Jacquet modules (and the Geometric Lemma, incorporated in the structure of a Hopf algebra). We use only some very basic general facts of the representation theory of reductive p-adic groups (the theory that we use was completed more then three decades ago, mainly in 1970-es). Of the specific results for general linear groups over A, basically we use only a very old result of G.I. Olshanskii, which says that there exist complementary series starting from $Ind(\rho\otimes\rho)$ whenever $\rho$ is a unitary irreducible cuspidal representation. In appendix of the paper "On parabolic induction on inner forms of the general linear group over a non-archimedean local field" of E. Lapid and A. Minguez, there is also a simple local proof of these results, based on a slightly different approach.

We show that the automorphism groups of countably categorical linear orders are extremely amenable. Using methods of Kechris, Pestov, and Todorcevic, we use this fact to derive a structural Ramsey theorem for certain families of finite ordered structures with finitely many partial equivalence relations with convex classes.

Let $G$ be a finite abelian group with exponent $n$, and let $r$ be a positive integer. Let $A$ be a $k\times m$ matrix with integer entries. We show that if $A$ satisfies some natural conditions and $|G|$ is large enough then, for each $r$--coloring of $G\setminus \{0\}$, there is $\delta$ depending only on $r,n$ and $m$ such that the homogeneous linear system $Ax=0$ has at least $\delta |G|^{m-k}$ monochromatic solutions. Density versions of this counting result are also addressed.

A century ago, Camille Jordan proved that the complex general linear group $GL_n(C)$ has the Jordan property: there is a Jordan constant $C_n$ such that every finite subgroup $H \le GL_n(C)$ has an abelian subgroup $H_1$ of index $[H : H_1] \le C_n$. We show that every connected algebraic group $G$ (which is not necessarily linear) has the Jordan property with the Jordan constant depending only on $\dim \, G$, and that the full automorphism group $Aut(X)$ of every projective variety $X$ has the Jordan property

Let $\F$ be a field with a non-trivial involution $c: \alpha \to \alpha^c$. An element $g \in {\rm GL}_n(\F)$ is called $c$-real if it is conjugate to $(g^c)^{-1}$. We prove that for $n \geq 2$, $g \in {\rm GL}_n(\F)$ is $c$-real if and only if it has a representation in some unitary group of degree $n$ over $\F$.

Let F be a finitely generated field of characteristic zero and \Gamma

We present the explicit global realization of the isometries of anti-de Sitter like spaces of signature $(d_-,d_+)$, and their algebras in the space of functions on the pseudo-Riemannian symmetric space $SO(d_- +1,d_+) / SO(d_-,d_+)$. The process of going to the invariant boundaries leads to the realization of the corresponding conformal groups and algebras. We compute systematically the geodesics in these spaces by considering the coset representation of them.

Let G be an algebraic group and let X be a smooth integral scheme over a field k. In this paper we construct homology-type groups $H_i(X,G)$ by considering cycles in the simplicial scheme $BG\times X (an idea suggested by Andrei Suslin). We discuss the basic properties of these groups and construct a spectral sequence, beginning with the groups $H_i(\Delta^j,G)$, which converges to the etale cohomology of the simplicial group BG. These groups are therefore connected with the study of Friedlander's generalized isomorphism conjecture. We also compute some examples, focusing in particular on the case X=Spec(k). In the case where k is the real numbers, there is a connection between the groups $H_i$ and the Z/2-equivariant cohomology of the classifying space of the discrete group $G(\mathbb R)$.

Fix a finite set $S \subset {GL}(k,\mathbb{Z})$. Denote by $a_n$ the number of products of matrices in $S$ of length $n$ that are equal to 1. We show that the sequence $\{a_n\}$ is not always P-recursive. This answers a question of Kontsevich.

In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H acting completely reducibly on a vector space V: if the orbits containing the vectors a and b have coprime lengths m and n, we prove that the orbit containing a+b has length mn. Such groups H are always reducible if n and m are greater than 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor. In the more general context of finite primitive permutation groups G, we show that coprime non-identity subdegrees are possible if and only if G is of O'Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for a finite primitive permutation group, a set of pairwise coprime subdegrees has size at most 2. Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.

In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H acting completely reducibly on a vector space V: if the orbits containing the vectors a and b have coprime lengths m and n, we prove that the orbit containing a+b has length mn. Such groups H are always reducible if n and m are greater than 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor. In the more general context of finite primitive permutation groups G, we show that coprime non-identity subdegrees are possible if and only if G is of O'Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for a finite primitive permutation group, a set of pairwise coprime subdegrees has size at most 2. Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.

We revisit the selection problem, namely that of computing the $i$th order statistic of $n$ given elements, in particular the classical deterministic algorithm by grouping and partition due to Blum, Floyd, Pratt, Rivest, and Tarjan~(1973). While the original algorithm uses groups of odd size at least $5$ and runs in linear time, it has been perpetuated in the literature that using groups of $3$ or $4$ will force the worst-case running time to become superlinear, namely $\Omega(n \log{n})$. We first point out that the arguments existent in the literature justifying the superlinear worst-case running time fall short of proving the claim. We further prove that it is possible to use group size $3$ or $4$ while maintaining the worst case linear running time. To this end we introduce two simple variants of the classical algorithm, the repeated step algorithm and the shifting target algorithm, both running in linear time.