## What is/are Linear Group?

Linear Group - The rings we encounter are graded rings of the form K [ U ] Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ.^{[1]}In this paper we study the construction on $\mathbb{F}_q^{2k}$ of full flag codes having maximum distance (optimum distance full flag codes) that can be endowed with an orbital structure provided by the action of a subgroup of the general linear group.

^{[2]}We prove that if a linear group Γ ⊂ GLn(K) over a field K of characteristic zero is boundedly generated by semi-simple (diagonalizable) elements then it is virtually solvable.

^{[3]}Finally, we investigate the Lie group symmetries, fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H3 and the matrix linear group SL(2,R) for the generalized Fokker-Plank equation (GFPE).

^{[4]}In this paper, we prove that projective special linear groups L3(q), where 0

^{[5]}Let ( g n ) n ⩾ 1 be a sequence of independent and identically distributed random elements of the general linear group G L d ( R ) , with law μ.^{[6]}Finally, we give an instantiation of leakage-resilient HPKE under the linear assumption in bilinear groups.^{[7]}Let G = S L ( 2 , 5 ) be the special linear group of 2 × 2 -matrices with coefficients in the field with 5 elements.^{[8]}Its novelty lies in the establishment of an optimization model with the nonlinear group self-esteem degree function as the objective function while group consensus threshold as the restrictions.^{[9]}The privacy of our construction relies on the SXDH assumption over bilinear groups via complexity leveraging.^{[10]}We formally prove that our scheme is secure also in case of an untrusted cloud server that colludes with a set of users, under the generic bilinear group model.^{[11]}A criterion for identifying the automorphism group as an algebraic subgroup the general linear group is given.^{[12]}The velocity of the soliton is close to the linear group velocity of ultrasound.^{[13]}In this note we review the development of the theory of rank for the case of the general linear group GLn over a finite field Fq, and give a proof of the ”agreement conjecture” that holds true for sufficiently large q.^{[14]}Spectral relaxation technique is used to acquire the numerical solution for the altered nonlinear group of differential equations.^{[15]}Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices.^{[16]}Our approach utilizes composite order bilinear group setting to achieve the tree based construction in public key setting.^{[17]}In order to address these problems, some ciphertext-policy attribute-based encryption (CP-ABE) schemes have been proposed to protect the privacy and security of data; but these schemes still have the following defects: 1) most of the existing hidden policy CP-ABE schemes only enable restricted access structure, such as “AND” gate; 2) several schemes supporting flexible access control are inefficient in decryption, because most of them are constructed in the composite order bilinear group; and 3) many of the proposed schemes fail to check the correctness of decryption message.^{[18]}In this paper, we propose a graph signature and proof system, where these are computed on bilinear groups without the RSA modulus.^{[19]}For the second fundamental representation of the general linear group over a commutative ring R, we construct straightforward and uniform polynomial expressions of elementary generators as products of elementary conjugates of an arbitrary matrix and its inverse.^{[20]}Flag codes defined as orbits of a cyclic subgroup of the general linear group acting on flags of Fq are called cyclic orbit flag codes.^{[21]}Also as expected, learning performance was found to depend on the type of performance curve with the best learning performance exhibited by the linear group.^{[22]}In this paper, we prove that there is a group homomorphism from general linear group over a polynomial extension of a local ring to the W.^{[23]}In this paper, we study the relation of the size of the class two quotients of a linear group and the size of the vector space.^{[24]}These lodes are spatially distributed as linear groups along the shear zone with distinct lithological assemblages.^{[25]}In this paper, we propose a novel privacy-preserving authentication protocol (P2BA) in bilinear groups, where a registered vehicle signs a traffic-related message and sends it to the nearby Road-side Unit (RSU) together with its blinded certificate.^{[26]}Since any invertible matrix maps subspaces to subspaces of the same dimension we have a group action of the general linear group, GL n ( F q ) , on G q ( k , n ) (Grassmann variety).^{[27]}The unexpected linear group 13 E[triple bond, length as m-dash]E triple bonds were herein uncovered with the D3h-symmetry E2M5+ (M = Li, Na, and K) clusters, where the linear M-E[triple bond, length as m-dash]E-M form is perfectly surrounded by M3 motifs.^{[28]}We show that if the Zariski closure of Γ=⟨S⟩\Gamma=\langle S\rangle is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay(Γ/Γ(q),πq(S))\operatorname{Cay}(\Gamma/\Gamma(q),\pi_{q}(S)) is O(logq)O(\log q), where 𝑞 is an arbitrary positive integer, πq:Γ→Γ/Γ(q)\pi_{q}\colon\Gamma\to\Gamma/\Gamma(q) is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.^{[29]}Based on the new authentication structure, we propose a public integrity auditing scheme, which is secure against forge attacks under the assumptions of the discrete logarithm problem and the computational Diffie-Hellman problem in bilinear groups in the random oracle model.^{[30]}The construction is built in (asymmetric) bilinear groups of prime order, and its unforgeability is derived in the standard model under (asymmetric version of) the well-studied decisional linear (DLIN) assumption coupled with the existence of standard collision resistant hash functions.^{[31]}We give a simple and unified proof that the unrestricted wreath product of a weak sofic, sofic, linear sofic or hyperlinear group by an amenable group is weak sofic, sofic, linear sofic or hyperlinear, respectively.^{[32]}We give an intrinsic criterion to tell whether a reflection factorization in the general linear group is reduced, and give a formula for computing reflection length in the general affine group.^{[33]}Motivated in parallel by questions in arithmetic and linear group theory, we classify all irreducible fixed-point subgroups of Sp_8(2) and give new infinite series of irreducible fixed-point subgroups of symplectic groups Sp_m(2) for various m arising from certain representations of groups of Lie type.^{[34]}) for projective special linear groups (2,41) and (2,43) find in this work to find the cyclic partition for each group.^{[35]}Finally, we analyze the security of PrivSTL against chosen-plaintext, chosen-keyword and outside keyword-guessing attacks in generic bilinear group model, and show that PrivSTL guarantees the spatio-temporal keyword profile privacy, and also protects the query privacy of mobile user.^{[36]}In this paper, we study the relations of the sizes of various sections of finite linear groups and the largest orbit size of the linear group actions.^{[37]}To date, a number of VLR-GS schemes have been proposed under bilinear groups and lattices, while they have not yet been instantiated based on coding theory.^{[38]}The proposed RS-HABE scheme features of forward security (a revoked user can no longer access previously encrypted data) and backward security (a revoked user also cannot access subsequently encrypted data) simultaneously, and is proved to be selectively secure under a complexity assumption in bilinear groups, without random oracles.^{[39]}The orientation-preserving combinatorial automorphism group of this regular map of Hurwitz is the projective special linear group PSL(2, 8).^{[40]}Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k.^{[41]}Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables with the degree of each [Formula: see text] being [Formula: see text] regarded as a module over the mod-[Formula: see text] Steenrod algebra [Formula: see text] and let [Formula: see text] be the general linear group over the prime field [Formula: see text] which acts naturally on [Formula: see text].^{[42]}However, the two schemes are not secure over the recommended linear group of Li et al.^{[43]}Therefore, we propose a policy-controlled signature scheme with strong expressiveness and privacy-preserving policy (PCS-PP), in which linear secret sharing schemes is to design access structure which has strong expression, the three primes composite order bilinear groups is used to hide the attribute value into the attribute name that may expose the privacy data by data distortion concept.^{[44]}This allows us to instantiate these primitives from various assumptions such as DDH or CDH (latter in bilinear groups), or the (R)LWE and the SIS assumptions.^{[45]}The construction is adaptively secure under standard k-Lin assumption in prime-order bilinear groups.^{[46]}We prove that the local Rankin–Selberg integrals for principal series representations of the general linear groups agree with certain simple integrals over the Rankin–Selberg subgroups, up to certain constants given by the local gamma factors.^{[47]}We present a construction of DFPE in prime-order bilinear groups, discuss a direct application of DPFE for enhancing security guarantees within Cloudflare’s Geo Key Manager, and show its generic use to construct forward-secret IBE and forward-secret digital signatures.^{[48]}In this paper, we study the existence of large orbits of odd-order subgroups of a finite solvable linear group G, and present some applications on bounding the odd-order parts of the chief series of an arbitrary linear group G.^{[49]}In this paper, we have computed the dimensions of the vector space of relative symmetric polynomials with respect to all irreducible characters of the projective special linear group PSL2(F7).^{[50]}## General Linear Group

The rings we encounter are graded rings of the form K [ U ] Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ.^{[1]}In this paper we study the construction on $\mathbb{F}_q^{2k}$ of full flag codes having maximum distance (optimum distance full flag codes) that can be endowed with an orbital structure provided by the action of a subgroup of the general linear group.^{[2]}Let ( g n ) n ⩾ 1 be a sequence of independent and identically distributed random elements of the general linear group G L d ( R ) , with law μ.^{[3]}A criterion for identifying the automorphism group as an algebraic subgroup the general linear group is given.^{[4]}In this note we review the development of the theory of rank for the case of the general linear group GLn over a finite field Fq, and give a proof of the ”agreement conjecture” that holds true for sufficiently large q.^{[5]}Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices.^{[6]}For the second fundamental representation of the general linear group over a commutative ring R, we construct straightforward and uniform polynomial expressions of elementary generators as products of elementary conjugates of an arbitrary matrix and its inverse.^{[7]}Flag codes defined as orbits of a cyclic subgroup of the general linear group acting on flags of Fq are called cyclic orbit flag codes.^{[8]}In this paper, we prove that there is a group homomorphism from general linear group over a polynomial extension of a local ring to the W.^{[9]}Since any invertible matrix maps subspaces to subspaces of the same dimension we have a group action of the general linear group, GL n ( F q ) , on G q ( k , n ) (Grassmann variety).^{[10]}We give an intrinsic criterion to tell whether a reflection factorization in the general linear group is reduced, and give a formula for computing reflection length in the general affine group.^{[11]}Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k.^{[12]}Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables with the degree of each [Formula: see text] being [Formula: see text] regarded as a module over the mod-[Formula: see text] Steenrod algebra [Formula: see text] and let [Formula: see text] be the general linear group over the prime field [Formula: see text] which acts naturally on [Formula: see text].^{[13]}We prove that the local Rankin–Selberg integrals for principal series representations of the general linear groups agree with certain simple integrals over the Rankin–Selberg subgroups, up to certain constants given by the local gamma factors.^{[14]}Finally, we apply our results to the invariant theory of tensor powers of the adjoint representation of the general linear group.^{[15]}Finally, we extend our result from the general linear group to both the orthogonal and symplectic groups.^{[16]}The Weyl group, which is the quotient of the normalizer by the centralizer, of the maximal elementary abelian subgroup E n in Spn is the general linear group GLn = GL(n,Fp ).^{[17]}Security of some present day public key cryptosystem (PKC) is based on general linear groups as it is a good choice for developing such types of cryptosystems.^{[18]}As applications, we give explicit presentations of a universal family of irreducible $p$-adic unitary Banach representations of the open unit disc of the general linear group and its $q$-deformation in the case of dimension $2$.^{[19]}This result is well-known in case of the action of the general linear group.^{[20]}This approach allowed us to study some algebraic representations of RS codes through the study of the general linear group GL(n,q).^{[21]}Let us recall the canonical generators of the general linear group GL2(Z), viz.^{[22]}First, we propose 16 different S-boxes based on projective general linear group and 16 primitive irreducible polynomials of Galois field of order 256, and then utilize these S-boxes with combination of chaotic map in image encryption scheme.^{[23]}These two categories are related via the exact forgetful functor The category is strongly related to topology and representation theory of symmetric and general linear groups but the homological algebra in is rather mysterious.^{[24]}The Kronecker coefficients are the structure constants for the restriction of irreducible representations of the general linear group GL(nm) into irreducibles for the subgroup GL(n) × GL(m).^{[25]}Likewise, will be proved that $\mathrm{H}^{\vee}$, has the decomposing in components as hyper-cohomology groups which can be characterized as H- states in Vec$_\mathbb{C}$, for field equations $d \textup{da}=0$, on the general linear group with $k=\mathbb{C}$.^{[26]}There is a particular tensor unfolding which gives rise to an isomorphism from this tensor space to the general linear group, i.^{[27]}2^{t+4}$ for all $t > 0$ and to describe the modular representations of the general linear group of rank 5 over $\mathbb F_2.^{[28]}We also prove similar result for the Steinberg group associated with any sufficiently isotropic general linear group constructed by a quasi-finite algebra.^{[29]}Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets.^{[30]}Let $GL_h$ be the usual general linear group of rank $h$ over $\mathbb Z/2.^{[31]}The projective general linear group $${\mathrm {PGL}}(2,q)$$ PGL ( 2 , q ) acts as a 3-transitive permutation group on the set of points of the projective line.^{[32]}For the infinite general linear group GL and X ∈ Sm(k), there is an isomorphism HomH•(k)(Σ m T Σ n SX+, BGL× Z) ∼= Kn−m(X) for n,m ≥ 0 [3, Theorem 3.^{[33]}By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition.^{[34]}Moreover, our method can be extended to general linear groups, and we prove that the lower bound of the sequential xor count based on words forMDS matrix over general linear groups is \begin{document}$ 4 \times 4 $\end{document} . \begin{document}$ 8n+2 $\end{document} ^{[35]}The article is about the representation theory of an inner form G of a general linear group over a non-Archimedean local field.^{[36]}The tensor product algebra TA(n) for the complex general linear group GL(n), introduced by Howe et al.^{[37]}In particular, in the case of the general linear group, it provides a geometric interpretation of the Robinson-Schensted correspondence between permutations and pairs of standard tableaux of the same shape.^{[38]}In the case of general linear groups and algebraic surfaces, we show that spectral surfaces admit canonical finite Cohen-Macaulayfications, which we call the Cohen-Macaulay spectral surfaces, and we use them to obtain a description of the generic fibers of the Hitchin morphism similar to the case of curves.^{[39]}I will consider random matrices in the general linear group GL(N;C) distributed according to a heat kernel measure.^{[40]}Let GLn be a complex general linear group.^{[41]}Fakta ini memotivasi suatu ide untuk menggantikan konsep bilangan bulat (integer) menjadi suatu matriks yang berukuran n n yang dinamakan dengana General Linear Group.^{[42]}We will give a definition of quadratic forms on bimodules and prove the sandwich classification theorem for subgroups of the general linear group $\mathrm{GL}(P)$ normalized by the elementary unitary group $\mathrm{EU}(P)$ if $P$ is a nondegenerate bimodule with large enough hyperbolic part.^{[43]}Let $G$ be the general linear group of the degree $n\geq 2$ over the field $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$.^{[44]}We are concerned with the irreducible, cuspidal representations of the general linear groups \(\mathrm {GL}(n, F)\).^{[45]}The (p-primary) equivariant reduced Euler characteristics of the building for the general linear group over a finite field are determined.^{[46]}The proofs involve methods and results of classical invariant theory, representation theory of the general linear group and explicit computations with matrices.^{[47]}These connect with various aspects of the representation theory of the symmetric groups and the general linear groups.^{[48]}This algebra describes, for a general linear group over a p-adic field, a large part of the unipotent block over fields of characteristic different from p.^{[49]}## Special Linear Group

In this paper, we prove that projective special linear groups L3(q), where 0^{[1]}Let G = S L ( 2 , 5 ) be the special linear group of 2 × 2 -matrices with coefficients in the field with 5 elements.^{[2]}We show that if the Zariski closure of Γ=⟨S⟩\Gamma=\langle S\rangle is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay(Γ/Γ(q),πq(S))\operatorname{Cay}(\Gamma/\Gamma(q),\pi_{q}(S)) is O(logq)O(\log q), where 𝑞 is an arbitrary positive integer, πq:Γ→Γ/Γ(q)\pi_{q}\colon\Gamma\to\Gamma/\Gamma(q) is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.^{[3]}) for projective special linear groups (2,41) and (2,43) find in this work to find the cyclic partition for each group.^{[4]}The orientation-preserving combinatorial automorphism group of this regular map of Hurwitz is the projective special linear group PSL(2, 8).^{[5]}In this paper, we have computed the dimensions of the vector space of relative symmetric polynomials with respect to all irreducible characters of the projective special linear group PSL2(F7).^{[6]}As the main theme, the explicit formulas for the number of spanning trees of commuting graphs associated with some specific groups, such as the Suzuki simple groups, the projective special linear groups, and some certain AC-groups, are obtained.^{[7]}The periodical slit and Artin indicator for the special linear group SL(2,49) from the induced character from the cyclic subgroups for the group after we calculate the irreducible rational representations of the group is computed in this work.^{[8]}We present a gauge formulation of the special affine algebra extended to include an antisymmetric tensorial generator belonging to the tensor representation of the special linear group.^{[9]}The polygon valued Minkowski valuations in R 2 intertwining with the special linear group are completely classified in this paper without any additional assumptions.^{[10]}In this paper, the author gives the discrete criteria and Jørgensen inequalities of subgroups for the special linear group on F̅((t)) in two and higher dimensions.^{[11]}Thus, we prove that, if an n -dimensional compact affine orbifold N is complete or if its holonomy group belongs to the special linear group S L ( n , R ) , then the Euler–Satake characteristic of N has to vanish.^{[12]}We show that any element of the special linear group $SL_2(R)$ is a product of two exponentials if the ring $R$ is either the ring of holomorphic functions on an open Riemann surface or the disc algebra.^{[13]}Let $G$ be the special linear group $\mathrm{SL}(2,q)$.^{[14]}These are special linear groups SL(d, q), orthogonal groups O(d, q), and symplectic groups Sp(d, q).^{[15]}## Unexpected Linear Group

The unexpected linear group 13 E[triple bond, length as m-dash]E triple bonds were herein uncovered with the D3h-symmetry E2M5+ (M = Li, Na, and K) clusters, where the linear M-E[triple bond, length as m-dash]E-M form is perfectly surrounded by M3 motifs.^{[1]}Strikingly, the unexpected linear group 13 E≡E triple bonds were herein found in the D4h -symmetry E2 Li6 2+ clusters, and they possess a large barrier (>18.^{[2]}## linear group gl

This approach allowed us to study some algebraic representations of RS codes through the study of the general linear group GL(n,q).^{[1]}The Kronecker coefficients are the structure constants for the restriction of irreducible representations of the general linear group GL(nm) into irreducibles for the subgroup GL(n) × GL(m).^{[2]}For the infinite general linear group GL and X ∈ Sm(k), there is an isomorphism HomH•(k)(Σ m T Σ n SX+, BGL× Z) ∼= Kn−m(X) for n,m ≥ 0 [3, Theorem 3.^{[3]}The tensor product algebra TA(n) for the complex general linear group GL(n), introduced by Howe et al.^{[4]}I will consider random matrices in the general linear group GL(N;C) distributed according to a heat kernel measure.^{[5]}## linear group model

We formally prove that our scheme is secure also in case of an untrusted cloud server that colludes with a set of users, under the generic bilinear group model.^{[1]}Finally, we analyze the security of PrivSTL against chosen-plaintext, chosen-keyword and outside keyword-guessing attacks in generic bilinear group model, and show that PrivSTL guarantees the spatio-temporal keyword profile privacy, and also protects the query privacy of mobile user.^{[2]}We then prove that the proposed ABKS-SM systems achieve selective security and resist off-line keyword-guessing attack in the generic bilinear group model.^{[3]}We prove the proposed signature algorithm is unforgeable against adaptively chosen messages attack in the generic bilinear group model under the continual leakage setting.^{[4]}## linear group setting

Our approach utilizes composite order bilinear group setting to achieve the tree based construction in public key setting.^{[1]}Recent series of works have shown that by limiting the message space of those schemes to the set of Diffie-Hellman (DH) pairs, it is possible to circumvent the known lower bounds in the Type-3 bilinear group setting thus obtaining the shortest signatures consisting of only 2 elements from the shorter source group.^{[2]}## linear group sl

Finally, we investigate the Lie group symmetries, fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H3 and the matrix linear group SL(2,R) for the generalized Fokker-Plank equation (GFPE).^{[1]}The periodical slit and Artin indicator for the special linear group SL(2,49) from the induced character from the cyclic subgroups for the group after we calculate the irreducible rational representations of the group is computed in this work.^{[2]}## linear group g

Let ( g n ) n ⩾ 1 be a sequence of independent and identically distributed random elements of the general linear group G L d ( R ) , with law μ.^{[1]}In this paper, we study the existence of large orbits of odd-order subgroups of a finite solvable linear group G, and present some applications on bounding the odd-order parts of the chief series of an arbitrary linear group G.^{[2]}## linear group gln

In this note we review the development of the theory of rank for the case of the general linear group GLn over a finite field Fq, and give a proof of the ”agreement conjecture” that holds true for sufficiently large q.^{[1]}The Weyl group, which is the quotient of the normalizer by the centralizer, of the maximal elementary abelian subgroup E n in Spn is the general linear group GLn = GL(n,Fp ).^{[2]}## linear group 13

The unexpected linear group 13 E[triple bond, length as m-dash]E triple bonds were herein uncovered with the D3h-symmetry E2M5+ (M = Li, Na, and K) clusters, where the linear M-E[triple bond, length as m-dash]E-M form is perfectly surrounded by M3 motifs.^{[1]}Strikingly, the unexpected linear group 13 E≡E triple bonds were herein found in the D4h -symmetry E2 Li6 2+ clusters, and they possess a large barrier (>18.^{[2]}## linear group psl

The orientation-preserving combinatorial automorphism group of this regular map of Hurwitz is the projective special linear group PSL(2, 8).^{[1]}If a two-dimensional projective linear group PSL (2,.^{[2]}