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Basic Algebra I

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... which can be handled in closed form through the Cardan solution [29]. In simulating the release process, it is assumed that release R t = R max , unless water in the reservoir is not sufficient, in which case R t = S t . ...
... The second type of benefit is associated with the reduction of CO 2 emissions. To assess this external effect, released energy was first turned into avoided tons of CO 2 using an emission coefficient of 433.2 CO 2 tons/GWh, according to Italian technical standard UNI/TS 11300-4; the economic value of avoided CO 2 tons was then attributed using the reference tables of the European Commission [29], which provide for each year of application (from 2010 to 2050) a range of values (lower-central-upper, in €/CO 2 tons) for the avoided emissions. For this application, the central value, spanning from 25 €/CO 2 ton in 2010 to 85 €/CO 2 ton in 2050, was used. ...
... The second type of benefit is associated with the reduction of CO2 emissions. To assess this external effect, released energy was first turned into avoided tons of CO2 using an emission coefficient of 433.2 CO2 tons/GWh, according to Italian technical standard UNI/TS 11300-4; the economic value of avoided CO2 tons was then attributed using the reference tables of the European Commission [29], which provide for each year of application (from 2010 to 2050) a range of values (lower-central-upper, in €/CO2 tons) for the avoided emissions. For this application, the central value, spanning from 25 €/CO2 ton in 2010 to 85 €/CO2 ton in 2050, was used. ...
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Pumped hydro storage (PHS) is one of the more suitable energy storage technologies to provide bulk storage of intermittent renewable energy sources (RES) such as wind. Since the main limiting factors to the expansion of this mature technology are environmental and financial concerns, the use of an existing reservoir can help mitigate both types of impacts. In addition, the high number of reservoirs for municipal and irrigation supply in many areas of the world makes the idea of using PHS as a relatively diffuse, open-market, technology for RES management attractive. These arguments in favor of PHS must, however, be convincing for investors and regulators from an economic standpoint. To this end, this paper presents a methodological tool to screen the feasibility of a PHS facility around an existing reservoir based on the principles of cost–benefit analysis, calibrated with data from Sicily, Italy. Each potential plant is characterized by two locational and two plant-specific parameters. Costs and benefits are assessed through a simulation model of the storage–release process on an hourly basis. Costs include both investment, and operation and maintenance expenditures, while the benefits considered include the opportunity cost of the current energy mix substituted by the stored energy, and the avoided CO2 emissions. The evaluation exercise is carried out parametrically, i.e., looking at a large number of combinations of the four parameters, in order to explore a wide range of possible plant configurations and to identify optimal ones under different locational conditions. A sensitivity analysis performed on models’ parameters points out the sensitivity of results to benefit, rather than cost-related, input parameters, such as the efficiency of the generating and pumping system and the opportunity cost of energy.
... Also do not form a group due a nonassociativide of multiplication. They form a Moufang Loop, a Loop with identity element (Jacobson [8]). ...
... Example 4. The Dirac delta (see [8]) can be poorly thought as a function on the real line which is zero everywhere except at the origin, where it is infinite, ...
... In order to generalize the Fourier transform to its octonionic form, first we described Fourier series and we introduced the hypercomplex model by considering results from [8,9,12], which are able to obtain a formulation for octonionic Fourier transform, which depends on an internal product. Furthermore, since not few models of Theoretical Physics may be analyzed through the geometry and algebra of hypercomplex, it will be our concern to concentrate the next steps in making all possible applications of our results in the context of unified physical theories for higher dimensional space-times. ...
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In this paper we present a construction of the Octonionic Fourier Series and we introduce a version for the Octonionic Fourier Transform with hypercomplex exponentials, besides we discuss a possible way of defining the convolution product for octonionic functions and also theoretical results. Through some examples, we illustrate the developed concepts for the octonionic transform and the convolution product.
... Before stating our structural description of D A (D), we need to recall some ideas associated with the Smith normal form of matrices over Z. The Smith normal form is a very useful part of the theory of integer matrices, and is discussed in many algebra texts, like [8,Chapter 3], [15, Chapter II] and [18,Chapter 9]. Definition 3 Suppose M is a matrix with κ columns and ρ rows, whose entries lie in a commutative ring with unity R. Then the elementary ideals ...
... Definition 3 reflects the indexing traditionally used in knot theory, according to which E j is generated by the determinants of the (κ−j)×(κ−j) submatrices. In algebra texts like [8,Chapter 3], [15, Chapter II] and [18,Chapter 9] it is common to refer instead to the jth ideal being generated by determinants of j × j submatrices. ...
... Because of our indexing convention we have φ j (M ) | φ j−1 (M ) ∀j, where many algebra texts would have the reverse. Also, our definition includes invariant factors equal to 0. Some references, like [18,Chapter 9], allow invariant factors equal to 0 but other references, like [8,Chapter 3] and [15, Chapter II], do not; instead they simply mention that the number of nonzero invariant factors is the rank of M over Q. ...
Article
If A is an abelian group and φ is an integer, let A(φ) be the subgroup of A consisting of elements a A such that φ ϵ a = 0. We prove that if D is a diagram of a classical link L and 0 = φ0,φ1,...,φn-1 are the invariant factors of an adjusted Goeritz matrix of D, then the group A(D) of Dehn colorings of D with values in A is isomorphic to the direct product of A and A = A(φ0),A(φ1),...,A(φn-1). It follows that the Dehn coloring groups of L are isomorphic to those of a connected sum of torus links T(2,φ1) # ⋯ # T(2,φn-1).
... Let (K, v, exp, D) be a differential valued exponential field, [8], with the domain E of the exponential function and the constant field C. Let x 1 , .., x n ∈ C and t ∈ K − C be such that tx 1 , ...., tx n ∈ E. Using the same argument in [6] (p. 278), and making obvious changes in appropriate places, one can obtain the following equivalent statements: ...
... Recall that Lindemann-Weierstrass Conjecture can be restated as follows: Let x 1,p , ..., x n,p be p− adic algebraic numbers over Q in the domain of exp p . If 1, exp p (x 1,p ), ..., exp p (x n,p ) are linearly dependent over Q, then not all of the elements x 1,p , ..., x n,p are distinct, see [6] (p. 278) (the proof in the p-adic setting still works with some slight modifications). ...
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In this paper we apply Ax-Schanuel's Theorem to the ultraproduct of the p−adic fields in order to prove a weak form of the p-adic Lindemann-Weierstrass conjecture for almost all primes.
... In Chapter 5 we prepare the mathematical ground for this ambitious goal, by studying the monoidal and lattice structures of families of compatible frames of discernment. We distinguish finite from general families of frames, describe the monoidal properties of compatible collections of both frames and refinings, and introduce the internal operation of 'maximal coarsening', which in turns induces in a family of frames the structures of Birkhoff, upper semimodular and lower semimodular lattice [567,568]. Both vector subspaces and families of frames share the structure of Birkhoff lattice [70] (Corollary 11). ...
... Two dual order relations. It is well-known (see [568], page 456) that the internal operation · of a monoid M induces an order relation (see Definition 36) | on the elements of M. Namely: ...
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In this Book we argue that the fruitful interaction of computer vision and belief calculus is capable of stimulating significant advances in both fields. From a methodological point of view, novel theoretical results concerning the geometric and algebraic properties of belief functions as mathematical objects are illustrated and discussed in Part II, with a focus on both a perspective 'geometric approach' to uncertainty and an algebraic solution to the issue of conflicting evidence. In Part III we show how these theoretical developments arise from important computer vision problems (such as articulated object tracking, data association and object pose estimation) to which, in turn, the evidential formalism is able to provide interesting new solutions. Finally, some initial steps towards a generalization of the notion of total probability to belief functions are taken, in the perspective of endowing the theory of evidence with a complete battery of estimation and inference tools to the benefit of all scientists and practitioners.
... 1. r is the product of all powers of irreducible factors in f which appear also in g with exponents more than or equal to the corresponding exponents in f , 2. s is the product of all powers of irreducible factors in f which appear also in g but with exponents strictly less than the corresponding exponents in f , 3. t is the product of all powers of irreducible factors in f which do not appear in g , 4. j is the product of all powers of irreducible factors in g which appear also in f with exponents less than or equal to the corresponding exponents in g , ...
... We use again Lemma 3 now with the polynomials f = min polw 2 = x 3 − 6x 2 + 11x − 6 and g = min polv 3 = x 2 − 2x + 1 to obtain a vector w 3 with minimum polynomial min polw 3 ...
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In this note, we give an easy algorithm to construct the rational canonical form of a square matrix or an endomorphism h of a finite dimensional vector space which does not depend on either the structure theorem for finitely generated modules over principal ideal domains or matrices over the polynomial ring. The algorithm is based on the construction of an element whose minimum polynomial coincides with the minimum polynomial of the endomorphism and on the fact that the h-invariant subspace generated by such an element admits an h-invariant complement. It is also shown that this element can be easily obtained without the factorisation of a polynomial as a product of irreducible polynomials.
... We also prove that a group is SSEP if its nonabelian subgroups are sufficiently large (proposition 1.1. 47), and that a metabelian group is SEP if its commutator subgroup has a certain structure (theorem 1.1.51). Proof. ...
... Gauss' beautiful proof, based on the lexicographic order on monomials, can be found in many sources, e.g. [47], Theorem 2.20, and [83], Theorem 1.1.1, and the original [37], paragraphs 3-5. ...
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This thesis addresses questions in representation and invariant theory of finite groups. The first concerns singularities of quotient spaces under actions of finite groups. We introduce a class of finite groups such that the quotients have at worst abelian quotient singularities. We prove that supersolvable groups belong to this class and show that nonabelian finite simple groups do not belong to it. The second question concerns the Cohen-Macaulayness of the invariant ring Z[x1,,xn]G\Z[x_1,\dots,x_n]^G, where G is a permutation group. We prove that this ring is Cohen-Macaulay if G is generated by transpositions, double transpositions, and 3-cycles, and conjecture that the converse is true as well.
... Note that the set Aut ± (g) = Aut(g) ∪ Aut − (g) is a group and it is called signed automorphisms group. Moreover, Aut(g) is of index two in Aut ± (g) and therefore is its normal subgroup [5]. ...
... Since Aut(sl 2 ) has only two cosets in LAut(sl 2 ) and a subgroup of index 2 is well-known to be normal [5], the following corollary is immediate. Remark 2.4 asserts that any anti-automorphism is a composition of one antiautomorphism with all automorphisms. ...
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We investigate local automorphisms of simple Lie algebra sln\mathfrak{sl}_n over a field of characteristic zero. In case n=2 we establish that the set of local automorphisms LAut(sl2)LAut(\mathfrak{sl}_2) is the group Aut±(sl2)Aut^{\pm}(\mathfrak{sl}_2) of all automorphisms and anti-automorphisms. For n3n\geq 3 we prove that Aut±(sln)Aut^{\pm}(\mathfrak{sl}_n) is a proper subgroup of LAut(sln)LAut(\mathfrak{sl}_n) of an infinite index.
... Summary. This is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/<p> as the desired field extension E [5], [3], [4]. ...
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This is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [ X ] \F there exists a field extension E of F such that p has a root over E . The formalization follows Kronecker’s classical proof using F [ X ] / as the desired field extension E [5], [3], [4]. In the first part we show that an irreducible polynomial p ∈ F [ X ] \F has a root over F [ X ] / . Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [ X ] / < p > as sets, so F is not a subfield of F [ X ] / , and hence formally p is not even a polynomial over F [ X ] / < p > . Consequently, we translate p along the canonical monomorphism ϕ : F → F [ X ] / and show that the translated polynomial ϕ ( p ) has a root over F [ X ] / . Because F is not a subfield of F [ X ] / we construct in this second part the field ( E \ ϕF )∪ F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [ X ] / and therefore consider F as a subfield of F [ X ] / ”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [ X ] = ∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑 2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [ X ] ≠ ∅. We also prove that for Mizar’s representations of 𝕑 n , 𝕈 and 𝕉 we have 𝕑 n ∩ 𝕑 n [ X ] = ∅, 𝕈 ∩ 𝕈 [ X ] = ∅ and 𝕉 ∩ 𝕉 [ X ] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E . Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E . We then apply the construction of the second part to F [ X ] / with the canonical monomorphism ϕ : F → F [ X ] / . Together with the first part this gives - for fields F with F ∩ F [ X ] = ∅ - a field extension E of F in which p ∈ F [ X ] \F has a root.
... Summary. This is the first part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/<p> as the desired field extension E [9], [4], [6]. ...
Article
Full-text available
This is the first part of a four-article series containing a Mizar [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/ p as the desired field extension E [5], [3], [4].
... see e. g. Jacobson [Jac96,p. 196]. The value a 1 is defined as the sum of all diagonal elements, thus the trace of U 2 k . ...
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The thesis is mainly about the construction and implementation of cyclic mutually unbiased bases, dealing with different entanglement structures by discussing the related group structures. A recursive construction for Fermat number dimensions is given and related to Wiedemann's conjecture for systems with more than 2048 qubits. The second part of the thesis analyses a quantum public-key encryption scheme.
... Summary. This is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/<p> as the desired field extension E [5], [3], [4]. ...
Preprint
Full-text available
This is the second part of a four-article series containing a Mizar formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p in F[X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using the field F[X]/ p as the desired field extension E.
... Summary. This is the first part of a four-article series containing a Mizar [2], [3] formalization of Kronecker's construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker's classical proof using F [X]/<p> as the desired field extension E [9], [4], [6]. ...
Preprint
Full-text available
This is the first part of a four-article series containing a Mizar formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F[X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using the field F[X]/ p as the desired field extension E.
... The term 'Nullstellensatz' is the name given to the following basic theorem (due to D. Hilbert) on the existence of zeros of ideals (see e.g. [7], volume II p.429) Theorem 11.5.3 (Nullstellensatz) If K is an algebraically closed field, A = K[X 1 , X 2 , . . . , X n ] a ring of polynomials with coefficients in K and I ⊆ A an ideal, then 1 ∈ I iff there is a point p = (a 1 , a 2 , . . . ...
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This book is a bridge between introductory books on topos theory such as Lawvere and Schanuel and full fledge topos books such as Mac Lane and Moerdijk.
... (i) In view of Theorem 3.12 and Proposition 2.5, we can observe that Lemma 3.11 generalizes the decomposition theory of comodules described by Xu-Shum-Fong in [11]. In addition, by Theorem 3.14, we can easily see that our Lemma 3.11 also covers the primary decomposition theory of finitely generated torsion modules over PID (see [5]). ...
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The decomposition theory of perfect modules is developed and consequently the decomposition of comodules is extended to perfect modules over an arbitrary ring. In addition, a decomposition theorem of finitely generated torsion modules over PID is established and the theory of Dickson on the primary decomposition of modules becomes a special case of our result.
... In this paper, we follow the notions and terminologies given in Jacobson [5] and Xu [8]. Our aim is to prove the fundamental theorem of Galois theory for a skew field F under the assumption that the group G of automorphisms of F is finite outer. ...
Article
In this paper,we study the Galois theory for the skew field F with a finite outer group G of automorphisms of F. For such an outer group G, we prove the Galois fundamental theorem which can be stated as the same as the classical Galois fundamental theorem for fields only by replacing the words "subfield" and "skew subfield". Moreover, we provide a new method to prove the Fundamental theorem for Galois corresponding subgroups to subrings of L(P, F). Although the Galois fundamental theorem for skew field was proved previously by Jacobson and Cohn in the literature, our approach is somewhat different, in particular , we do not use any field theory but only use the techniques of complete rings of linear transformations developed by Xu in 1980.
... In this paper, we follow the notions and terminologies given in Jacobson [5] and Xu [8]. Our aim is to prove the fundamental theorem of Galois theory for a skew field F under the assumption that the group G of automorphisms of F is finite outer. ...
Article
In this paper,we study the Galois theory for the skew field F with a finite outer group G of automorphisms of F. For such an outer group G, we prove the Galois fundamental theorem which can be stated as the same as the classical Galois fundamental theorem for fields only by replacing the words "subfield" and "skew subfield". Moreover, we provide a new method to prove the Fundamental theorem for Galois corresponding subgroups to subrings of L(P, F). Although the Galois fundamental theorem for skew field was proved previously by Jacobson and Cohn in the literature, our approach is somewhat different, in particular , we do not use any field theory but only use the techniques of complete rings of linear transformations developed by Xu in 1980.
... (i) In view of Theorem 3.12 and Proposition 2.5, we can observe that Lemma 3.11 generalizes the decomposition theory of comodules described by Xu-Shum-Fong in [11]. In addition, by Theorem 3.14, we can easily see that our Lemma 3.11 also covers the primary decomposition theory of finitely generated torsion modules over PID (see [5]). ...
... The condition (5. 16) in [25] was |a n | ≥ ( √ 2|a n−1 |) τ , ∀n ≥ 1. ...
Thesis
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This work is devoted to study the algebraic independence in the p− adic Complex field Cp . In particular, we discuss the algebraic independence between the p− adic exponentials and their arguments.
... The computation of the non-stochastic and stochastic bilinear indices is develop by using the k th "nonstochastic and stochastic graph-theoretical electronic-density matrices" M k and S k , correspondingly, as matrices of the mathematical forms. [28][29][30] These matricial operators are graph-theoretical electronic-structure models, like the ''extended Hückel MO model.'' The M 1 matrix considers all valence-bond electrons (σ-and π-networks) in one step, and their power k (k = 0, 1, 2, 3,...) can be considered as an interactingelectronic chemical-network in step k. ...
... (i) In view of Theorem 3.12 and Proposition 2.5, we can observe that Lemma 3.11 generalizes the decomposition theory of comodules described by Xu-Shum-Fong in [11]. In addition, by Theorem 3.14, we can easily see that our Lemma 3.11 also covers the primary decomposition theory of finitely generated torsion modules over PID (see [5]). ...
Article
The decomposition theory of perfect modules is developed and consequently the decomposition of comodules is extended to perfect modules over an arbitrary ring. In addition, a decomposition theorem of finitely generated torsion modules over PID is established and the theory of Dickson on the primary decomposition of modules becomes a special case of our result.
... (The irreducibility of N follows from the fact that its restriction on M 3 (k) is the usual determinant, which is irreducible, as shown in e.g. [J2,Theorem 7.2].) By [ALM], the stabilizer of 1 is Aut(A), which is connected. ...
Preprint
We study Albert algebras and related affine group schemes over rings from the point of view of torsors and cohomology. We begin by showing that isotopes of Albert algebras are obtained as twists by a certain F4\mathrm F_4-torsor with total space a group of type E6\mathrm E_6. We then consider certain D4\mathrm D_4-torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalized reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is non-trivial, we conclude that the Albert algebra does not uniquely determine the underlying composition. Finally, we strengthen this result by showing that a given reduced Albert algebra can admit two coordinate algebras which are non-isomorphic and have non-isometric quadratic forms, contrary, in a strong sense, to the case over fields, as established by Albert and Jacobson.
... Proofs can be found in [4] and [6]. We can get the total number of real roots by looking at the limits as a → −∞ and b → +∞. ...
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For the general monic quintic with real coefficients, polynomial conditions on the coefficients are derived as directly and as simply as possible from the Sturm sequence that will determine the real and complex root multiplicities together with the order of the real roots with respect to multiplicity.
... Proof. This follows from the standard Second Isomorphism Theorem for algebraic structures (e.g., see [28]). ...
... In this paper, we follow the notions and terminologies given in Jacobson [5] and Xu [8]. Our aim is to prove the fundamental theorem of Galois theory for a skew field F under the assumption that the group G of automorphisms of F is finite outer. ...
Article
In this paper,we study the Galois theory for the skew field F with a finite outer group G of automorphisms of F. For such an outer group G, we prove the Galois fundamental theorem which can be stated as the same as the classical Galois fundamental theorem for fields only by replacing the words "subfield" and "skew subfield". Moreover, we provide a new method to prove the Fundamental theorem for Galois corresponding subgroups to subrings of L(P, F). Although the Galois fundamental theorem for skew field was proved previously by Jacobson and Cohn in the literature, our approach is somewhat different, in particular , we do not use any field theory but only use the techniques of complete rings of linear transformations developed by Xu in 1980.
... In this paper, we follow the notions and terminologies given in Jacobson [5] and Xu [8]. Our aim is to prove the fundamental theorem of Galois theory for a skew field F under the assumption that the group G of automorphisms of F is finite outer. ...
Article
In this paper,we study the Galois theory for the skew field F with a finite outer group G of automorphisms of F. For such an outer group G, we prove the Galois fundamental theorem which can be stated as the same as the classical Galois fundamental theorem for fields only by replacing the words "subfield" and "skew subfield". Moreover, we provide a new method to prove the Fundamental theorem for Galois corresponding subgroups to subrings of L(P, F). Although the Galois fundamental theorem for skew field was proved previously by Jacobson and Cohn in the literature, our approach is somewhat different, in particular , we do not use any field theory but only use the techniques of complete rings of linear transformations developed by Xu in 1980.
... Most of the mathematics encountered in this paper can be found in standard texts such as [20,21,11,7]. Our proof Cauchy-Schwarz is similar to the one in [19,Chapter 9] and further reading specific to relevant inequalities can be found in [22]. ...
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We present a mechanical proof of the Cauchy-Schwarz inequality in ACL2(r) and a formalisation of the necessary mathematics to undertake such a proof. This includes the formalisation of R^n as an inner product space. We also provide an application of Cauchy-Schwarz by formalising R^n as a metric space and exhibiting continuity for some simple functions R^n –> R. The Cauchy-Schwarz inequality relates the magnitude of a vector to its projection (or inner product) with another:
... Further background on non-standard analysis can be found in [12,1,10]. Background on vector, inner product, and metric spaces can be found in [15,13,14,9,6]. We also outline some theorems involving convex functions. Standard texts on convex optimisation include [2,11]. ...
Article
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This paper builds upon our prior formalisation of R^n in ACL2(r) by presenting a set of theorems for reasoning about convex functions. This is a demonstration of the higher-dimensional analytical reasoning possible in our metric space formalisation of R^n. Among the introduced theorems is a set of equivalent conditions for convex functions with Lipschitz continuous gradients from Yurii Nesterov's classic text on convex optimisation. To the best of our knowledge a full proof of the theorem has yet to be published in a single piece of literature. We also explore "proof engineering" issues, such as how to state Nesterov's theorem in a manner that is both clear and useful.
... That U → U * is an order-reversing involution whenever b is non-degenerate is shown in [Jac85, , together with the fact that ...
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In \cite{rump_goml}, Rump defined and characterized noncommutative universal groups G(X) for generalized orthomodular lattices X. We give an explicit description of G(X) in terms of \emph{paraunitary} matrix groups, whenever X is the orthomodular lattice of subspaces of a finite-dimensional k-vector space V that is equipped with an anisotropic, symmetric k-bilinear form.
... The theory of the Smith normal form is explained in many algebra books, like [10,15]. We summarize the ideas briefly. ...
Preprint
We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module M over the Laurent polynomial ring Λμ=Z[t1±1,,tμ±1]\Lambda_{\mu}=\mathbb{Z}[t_1^{\pm1},\dots,t_{\mu}^{\pm1}]. If D is a diagram of a link L then the colorings of D with values in M form a Λμ\Lambda_{\mu}-module ColorA(D,M)\mathrm{Color}_A(D,M). Extending a result of Inoue [Kodai Math.\ J.\ 33 (2010), 116-122], we show that ColorA(D,M)\mathrm{Color}_A(D,M) is isomorphic to the module of Λμ\Lambda_{\mu}-linear maps from the Alexander module of L to M. In particular, suppose M=F is a field, considered as a Λμ\Lambda_{\mu}-module via a ring homomorphism φ:ΛμF\varphi:\Lambda_{\mu} \to F. Then ColorA(D,M)\mathrm{Color}_A(D,M) is a vector space over F, and we show that its dimension is determined by the images under φ\varphi of the elementary ideals of L. This result applies in the special case of Fox tricolorings, which correspond to M=GF(3) and φ(ti)1\varphi(t_i) \equiv-1. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine ColorA(D,M)|\mathrm{Color}_A(D,M)|; this observation corrects erroneous statements of Inoue [J. Knot Theory Ramifications 10 (2001), 813-821; op. cit.].
... We say that two extensions {e} → A Lemma A. 9 If ω 1 , ω 2 ∈ Z 2 eq (G, A) and ω 2 −ω 1 ∈ B 2 eq (G, A), then [A× ω 1 G] = [A × ω 2 G]. ...
Article
A locally compact contraction group is a pair (G,f) where G is a locally compact group and f an automorphism of G which is contractive in the sense that the forward orbit under f of each g in G converges to the neutral element e, as n tends to infinity. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups (G,f) which are central extensions of the additive group of the field of formal Laurent series over Z/pZ by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.
... See, e.g.,[40, II,Prop. 3.5] for various left module structures on the Hom-space. ...
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We construct a new basis for a slim cyclotomic q-Schur algebra \cysSr via symmetric polynomials in Jucys--Murphy operators of the cyclotomic Hecke algebra \cysHr. We show that this basis, labelled by matrices, is not the double coset basis when \cysHr is the Hecke algebra of a Coxeter group, but coincides with the double coset basis for the corresponding group algebra, the Hecke algebra at q=1. As further applications, we then discuss the cyclotomic Schur--Weyl duality at the integral level. This also includes a category equivalence and a classification of simple objects.
... For convenience we choose h 1 = e. A careful discussion of the existence of a normal basis element is given in section 14.14 of Jacobson's book, Basic Algebra I [3]. It is proven there that γ is a normal basis element if and only if det(h i h j (γ)) = 0. We will use this important fact later. ...
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An important theorem of C. Hermite asserts that any set of algebraic number fields, whose discriminants are bounded in absolute value, must be finite. Properly formulated, a similar theorem holds true for function fields in one variable over a finite constant field. This paper gives a new proof of this result by using an analogue of the geometry of numbers approach due to H. Minkowski in the number field case. © Société Arithmétique de Bordeaux, 2017, tous droits réservés.
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... (i): This follows from Dedekind's Independence Theorem, see e.g. [J85,Sec. 4.14]. ...
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