Norm Principle Research Articles

Weak approximation on the norm one torus

For any abelian group $A$ , we prove an asymptotic formula for the number of $A$ -extensions $K/\mathbb {Q}$ of bounded discriminant such that the associated norm one torus $R_{K/\mathbb {Q}}^1 \mathbb {G}_m$ satisfies weak approximation. We are also able to produce new results on the Hasse norm principle and to provide new explicit values for the leading constant in some instances of Malle's conjecture.

A note on the Hasse norm principle

AbstractLet be a finite, abelian group. We show that the density of ‐extensions satisfying the Hasse norm principle exists, when the extensions are ordered by discriminant. This strengthens earlier work of Frei–Loughran–Newton, who obtained a density result under the additional assumption that is cyclic with denoting the smallest prime divisor of .

Analysis of reaction forces in fixture locating points: An Analytical, numerical, and experimental study

Reaction forces are important parameters in fixture design. They are generated by the clamping forces and machining loads at the fixture locating points. These forces are used as input values in the determination of clamping forces, fixture stiffness, and workpiece deformation. In this paper, an analytical model based on the minimum norm principle was developed to calculate these forces. Numerical simulations and experimental tests were performed on a 3D polyhedral workpiece to validate the model. The simulations were conducted using Abaqus® software and the experimental tests used a fixture and a 3D polyhedral workpiece. The theoretical, numerical, and experimental results showed good agreement for the normal component of reaction forces. The maximum errors of 3.9% and 15% were observed between the theoretical predictions compared to the numerical and experimental results, respectively. The model was also used to study the effects of two influential parameters, the coefficient of friction and clamping force, on the reaction forces. The good agreement between the theoretical, numerical, and experimental results demonstrated the efficiency of the proposed model in the rapid calculation of reaction forces for fixturing 3D polyhedral workpieces.

Norm Principle for Even K-groups of Number Fields

We investigate the norm maps of algebraic even K-groups of finite extensions of number fields. Namely, we show that they are surjective in most situations. In the event that they are not surjective, we give a criterion in determining when an element in the even K-group of the base field comes from a norm of an element from the even K-groups of the extension field. This latter criterion is only reliant on the real primes of the base field.

Norm one tori and Hasse norm principle, II: Degree 12 case

Let k be a field, T be an algebraic k-torus, X be a smooth k-compactification of T and PicX‾ be the Picard group of X‾=X×kk‾. Hoshi, Kanai and Yamasaki [HKY22] determined H1(k,PicX‾) for norm one tori T=RK/k(1)(Gm) and gave a necessary and sufficient condition for the Hasse norm principle for extensions K/k of number fields with [K:k]=n≤15 and n≠12. In this paper, we determine 64 cases with H1(k,PicX‾)≠0 and give a necessary and sufficient condition for the Hasse norm principle for K/k with [K:k]=12.

Norm one Tori and Hasse norm principle

Let k k be a field and T T be an algebraic k k -torus. In 1969, over a global field k k , Voskresenskiǐ proved that there exists an exact sequence 0 → A ( T ) → H 1 ( k , Pic ⁡ X ¯ ) ∨ → Ш ( T ) → 0 0\to A(T)\to H^1(k,\operatorname {Pic}\overline {X})^\vee \to \Sha (T)\to 0 where A ( T ) A(T) is the kernel of the weak approximation of T T , Ш ( T ) \Sha (T) is the Shafarevich-Tate group of T T , X X is a smooth k k -compactification of T T , X ¯ = X × k k ¯ \overline {X}=X\times _k\overline {k} , Pic ⁡ X ¯ \operatorname {Pic}\overline {X} is the Picard group of X ¯ \overline {X} and ∨ \vee stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus T = R K / k ( 1 ) ( G m ) T=R^{(1)}_{K/k}(\mathbb {G}_m) of K / k K/k , Ш ( T ) = 0 \Sha (T)=0 if and only if the Hasse norm principle holds for K / k K/k . First, we determine H 1 ( k , Pic ⁡ X ¯ ) H^1(k,\operatorname {Pic} \overline {X}) for algebraic k k -tori T T up to dimension 5 5 . Second, we determine H 1 ( k , Pic ⁡ X ¯ ) H^1(k,\operatorname {Pic} \overline {X}) for norm one tori T = R K / k ( 1 ) ( G m ) T=R^{(1)}_{K/k}(\mathbb {G}_m) with [ K : k ] = n ≤ 15 [K:k]=n\leq 15 and n ≠ 12 n\neq 12 . We also show that H 1 ( k , Pic ⁡ X ¯ ) = 0 H^1(k,\operatorname {Pic} \overline {X})=0 for T = R K / k ( 1 ) ( G m ) T=R^{(1)}_{K/k}(\mathbb {G}_m) when the Galois group of the Galois closure of K / k K/k is the Mathieu group M n ≤ S n M_n\leq S_n with n = 11 , 12 , 22 , 23 , 24 n=11,12,22,23,24 . Third, we give a necessary and sufficient condition for the Hasse norm principle for K / k K/k with [ K : k ] = n ≤ 15 [K:k]=n\leq 15 and n ≠ 12 n\neq 12 . As applications of the results, we get the group T ( k ) / R T(k)/R of R R -equivalence classes over a local field k k via Colliot-Thélène and Sansuc’s formula and the Tamagawa number τ ( T ) \tau (T) over a number field k k via Ono’s formula τ ( T ) = | H 1 ( k , T ^ ) | / | Ш ( T ) | \tau (T)=|H^1(k,\widehat {T})|/|\Sha (T)| .

Number fields with prescribed norms (with an appendix by Yonatan Harpaz and Olivier Wittenberg)

We study the distribution of extensions of a number field k with fixed abelian Galois group G , from which a given finite set of elements of k are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for 100% of G -extensions of k , when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.

"On the maximum modulus principle and the identity theorem in arbitrary dimension"

"We prove an identity theorem for Gˆateaux holomorphic functions on polygonally connected 2- open sets, which yields a very general maximum norm principle and a sublinear “max-min” principle. All results apply in particular to vector-valued functions which are holomorphic (in any sense that implies Gˆateaux holomorphy) on domains in Hausdorff locally convex spaces."

Constructing abelian extensions with prescribed norms

Given a number field K K , a finite abelian group G G and finitely many elements α 1 , … , α t ∈ K \alpha _1,\ldots ,\alpha _t\in K , we construct abelian extensions L / K L/K with Galois group G G that realise all of the elements α 1 , … , α t \alpha _1,\ldots ,\alpha _t as norms of elements in L L . In particular, this shows existence of such extensions for any given parameters. Our approach relies on class field theory and a recent formulation of Tate’s characterisation of the Hasse norm principle, a local-global principle for norms. The constructions are sufficiently explicit to be implemented on a computer, and we illustrate them with concrete examples.

Explicit methods for the Hasse norm principle and applications toAnandSnextensions

AbstractLetK/kbe an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle forK/kand the defect of weak approximation for the norm one torus\[R_{K/k}^1{\mathbb{G}_m}\]. We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure ofK/khas symmetric or alternating Galois group.

On the Tits–Weiss conjecture and the Kneser–Tits conjecture for and (With an Appendix by R. M. Weiss)

AbstractWe prove that the structure group of any Albert algebra over an arbitrary field isR-trivial. This implies the Tits–Weiss conjecture for Albert algebras and the Kneser–Tits conjecture for isotropic groups of type$\mathrm {E}_{7,1}^{78}, \mathrm {E}_{8,2}^{78}$. As a further corollary, we show that some standard conjectures on the groups ofR-equivalence classes in algebraic groups and the norm principle are true for strongly inner forms of type$^1\mathrm {E}_6$.

On the norm and multiplication principles for norm varieties

Let p be a prime, and suppose that F is a field of characteristic zero which is p-special (that is, every finite field extension of F has dimension a power of p). Let α∈𝒦nM(F)∕p be a nonzero symbol and X∕F a norm variety for α. We show that X has a 𝒦mM-norm principle for any m, extending the known 𝒦1M-norm principle. As a corollary we get an improved description of the kernel of multiplication by a symbol. We also give a new proof for the norm principle for division algebras over p-special fields by proving a decomposition theorem for polynomials over F-central division algebras. Finally, for p=n=m=2 we show that the known 𝒦1M-multiplication principle cannot be extended to a 𝒦2M-multiplication principle for X.

Modeling of jamming phenomenon in fixture design application: an analytical, numerical, and experimental study

Jamming may occur during the loading of the workpiece in the fixture and cause an improper locating process or damage to the workpiece, fixture, and loading system. Consideration of the jamming phenomenon is a necessary task in the planning of the workpiece loading strategy. In the present paper, the minimum norm principle is used to present an analytical model for predicting the jamming occurrence conditions through calculating reaction forces at the contact points. The implementation of this model results in an optimization problem that is solved using the augmented Lagrange multiplier method. The proposed model is applied to two well-known jamming problems, namely the peg-in-hole and block and palm configurations. Numerical analysis is also conducted in Adams software to verify the theoretical predictions. Finally, the experimental setups are fabricated for these configurations to validate the theoretical predictions and numerical results. The results show that the maximum errors between the theoretical predictions and experimental results are 14.4% and 3.8% for the jamming-out angle in the peg-in-hole study and jamming-in travel in the block and palm problem, respectively. Moreover, the worst-case error of 6.9% is obtained for the theoretical prediction of the jamming-in travel of block compared to the numerical result. Agreement between the theoretical, numerical, and experimental results proves the capabilities of the suggested model in the accurate prediction of the jamming occurrence conditions and its potentials for use in the three-dimensional jamming-prone fixturing systems.

Similarity of quadratic and symmetric bilinear forms in characteristic 2

We say that a field extension L/F has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over F that become isometric (resp. similar) over L are already isometric (resp. similar) over F. The famous Artin–Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic 2. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic 2, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau’s norm principle for arbitrary quadratic or bilinear forms in characteristic 2.

Maximum Principles for Matrix-Valued Analytic Functions

To what extent is the maximum modulus principle for scalar-valued analytic functions valid for matrix-valued analytic functions? In response, we discuss some maximum norm principles for such functions that do not appear to be widely known, deduce maximum and minimum principles for their singular values, and make some observations concerning resolvents and matrix exponentials.

The Hasse norm principle for An-extensions

We prove that, for every n≥5, the Hasse norm principle holds for a degree n extension K/k of number fields with normal closure N such that Gal(N/k)≅An. We also show the validity of weak approximation for the associated norm one tori.

Hasse principles for multinorm equations

A classical result of Hasse states that the norm principle holds for finite cyclic extensions of global fields, in other words local norms are global norms. We investigate the norm principle for finite dimensional commutative étale algebras over global fields; since such an algebra is a product of separable extensions, this is often called the multinorm principle. Under the assumption that the étale algebra contains a cyclic factor, we give a necessary and sufficient condition for the Hasse principle to hold, in terms of an explicitly constructed element of a finite abelian group. This can be seen as an explicit description of the Brauer-Manin obstruction to the Hasse principle.

The norm principle for type $D_n$ groups over complete discretely valued fields

The norm principle for type $D_n$ groups over complete discretely valued fields

H∞ norm principle in residual filter design for discrete-time linear positive systems

New method of residual filter synthesis for discrete-time linear strictly positive systems is developed in this paper. Introducing constraints of positive system parameter structure in the set of linear matrix inequalities and completing it with the inequality guaranteing asymptotic stability of the discrete-time positive linear observer structure, the formulation provides a generalized solution by which the observers and residual filters can be designed for positive and strictly positive systems. Capabilities of the proposed design conditions are demonstrated using numerical illustrative examples.

Konsepsi Penyelamatan Dana Desa Dari Perbuatan Korupsi

This paper aims to analyze the management of village funds from corruption. The problem focuses on how the concept of saving village funds from corruption? This research is in the form of normative legal research with approach of legal norm and legal principles. The result of the research shows that the concept of fund grant from villages from corruption bundles, namely, First, MoU with community with the aim of committing to build village together with village fund monitoring team; Second, establishing an independent team of supervisors to oversee the running of village fund management processes; third, ready to be sworn the village apparatus in the oath by using the scriptures of each religion; four strict sanctions with a view to providing perpetrators of village funds.