First Order Extensions of Residue Classes and Uniform Circuit Complexity

Separations among the first order logic Ring(0,+, ∗) of finite residue class rings, its extensions with generalized quantifiers, and in the presence of a built-in order are shown, using algebraic methods from class field theory. These methods include classification of spectra of sentences over finite residue classes as systems of congruences, and the study of their h-densities over the set of all prime numbers, for various functions h on the natural numbers. Over ordered structures the logic of finite residue class rings and extensions are known to capture DLOGTIMEuniform circuit complexity classes ranging from AC to TC. Separating these circuit complexity classes is directly related to classifying the h-density of spectra of sentences in the corresponding logics of finite residue classes. We further give general conditions under which a logic over the finite residue class rings has a sentence whose spectrum has no h-density. One application of this result is that Supported by MICINN project TIN2011-27479-C04-03 (BASMATI), MINECO project TIN201457226-P (APCOM) and Gen. Cat. project SGR2014-890 (MACDA).

FROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The Hilbert Class Field and p = x 2 + ny 2 . The Hilbert Class Field and Genus Theory. Orders in Imaginary Quadratic Fields. Class Fields Theory and the Cebotarev Density Theorem. Ring Class Field and p = x 2 + ny 2 . COMPLEX MULTIPLICATION. Elliptic Functions and Complex Multiplication. Modular Functions and Ring Class Fields. Modular Functions and Singular j--Invariants. The Class Equation. Ellpitic Curves. References. Index.

On the Circuit Complexity of Sigmoid Feedforward Neural Networks

The universal binary neurons are a central point of this Chapter. The mathematical models of the universal binary neuron over the field of the complex numbers, the residue class ring and the finite field are considered. A notion of the P-realizable Boolean function, which may be considered as a generalization of a notion of the threshold Boolean function, is introduced. It is shown that the implementation with a single neuron the input/output mapping described by non-threshold Boolean functions is possible, if weights are complex, and activation function of the neuron is a function of the argument of the weighted sum (similar to multi-valued neuron). It is also possible to define an activation function on the residue class ring and the finite field in such a way that the implementation of non-threshold Boolean functions will be possible on the single neuron with weights from these sets. P-realization of multiple-valued function over finite algebras and residue class ring in particular is also considered. The general features of P-realizable Boolean functions are considered, also as the synthesis of universal binary neuron.

Olga Taussky’s interest in class field theory began when she was a student with Ph. Furtwangler at the University of Vienna (she received her doctorate under his supervision in 1930). At this time, E. Artin, using his general reciprocity law, reformulated the “Principal Ideal Theorem” as a purely grouptheoretic question regarding the triviality of the “Verlagerung” homomorphism from a finite group to its commutator subgroup. Since Furtwangler had been developing group theoretic methods in his own number theoretic investigations, Artin communicated this new idea to him. This was considered to be one of the central problems in the subject at the time, and so there was a good deal of excitement when Furtwangler eventually succeeded in proving the result using this new approach. Olga Taussky had been looking for a thesis topic and Furtwangler suggested several questions related to the finer structure of the Principal Ideal Theorem. The optimistic atmosphere, and the prospect of contributing at the frontiers of class field theory, sparked in Olga a deep interest in this problem which lasted her entire career. She returned to the questions of “capitulation”, a term coined by one of her co-authors Arnold Scholz, several times in her life, always with the sense that these were questions of deep arithmetic significance. I believe that it was the experience of her early contact with class field theory which dominated her research interests in number theory and which provided her a fertile source of inspiration for most of her career. I will describe some of her work in this area below. I would like to take this opportunity to thank the referee for many valuable suggestions. For a number field F , (i.e., an extension of finite degree over the rational field Q) its Hilbert class field H = H(F ) is the maximal Galois extension of F which is everywhere unramified and whose Galois group Gal(H/F ) is abelian. It is a consequence of Artin’s reciprocity law that Gal(H/F ) is isomorphic to the ideal class group C(F ) of F. The Principal Ideal Theorem is the statement that every ideal of F becomes a principal ideal when considered

Any single system whose space of states is given by a separable Hilbert space is automatically equipped with infinitely many hidden tensor-like structures. This includes all quantum mechanical systems as well as classical field theories and classical signal analysis. Accordingly, systems as simple as a single one-dimensional harmonic oscillator, an infinite potential well, or a classical finite-amplitude signal of finite duration can be decomposed into an arbitrary number of subsystems. The resulting structure is rich enough to enable quantum computation, violation of Bell’s inequalities, and formulation of universal quantum gates. Less standard quantum applications involve a distinction between position and hidden position. The hidden position can be accompanied by a hidden spin, even if the particle is spinless. Hidden degrees of freedom are, in many respects, analogous to modular variables. Moreover, it is shown that these hidden structures are at the roots of some well-known theoretical constructions, such as the Brandt–Greenberg multi-boson representation of creation–annihilation operators, intensively investigated in the context of higher-order or fractional-order squeezing. In the context of classical signal analysis, the discussed structures explain why it is possible to emulate a quantum computer by classical analog circuit devices.

The authors describe mapping, scaling, and conversion processes using a new mapping strategy for the modulus replication residue number system (MRRNS). The strategy allows direct mapping of bits of either a purely real or multiplexed bit coded complex number to a set of independent rings, defined by moduli 3, 5, and 7. The MRRNS technique is superior to a large QRNS system operating with a computational dynamic range of over 27 b. A classical radix-4 implementation of a 1024 FFT is used for the comparison. The scaling and conversion procedure is shown to be a set of finite ring calculations followed by an array of ordinary binary adders. The VLSI implementation of the most complex finite ring circuit required (a Mod 7 multiplier) is shown to be easily implemented using the switching tree approach, and mask extracted simulations at 50 MHz demonstrate the embedding of the switching trees in a dynamic pipeline/evaluate circuit with restoring latch.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On-shell constructibility is redefining our understanding of perturbative quantum field theory. The tree-level S-matrix of constructible theories is completely determined by a set of recurrence relations and a reduced number of scattering amplitudes. In this paper, we revisit the on-shell constructibility of gravitational theories making use of new results on soft theorems and recurrence relations. We show that using a double complex shift and an all-line soft deformation allows us to relax the technical conditions for constructibility, in order to include more general propagators and higher-derivative interactions that prevent using conventional Britto-Cachazo-Feng-Witten (BCFW) shifts. From this result we extract a set of criteria that guarantee that a given gravitational action has the same tree-level S-matrix in Minkowski spacetime as general relativity, which implies the equivalence at all orders in perturbation theory between these classical field theories on asymptotically flat spacetimes. As a corollary we deduce that the scattering amplitudes of general relativity and unimodular gravity are the same for an arbitrary number of external particles (as long as the S-matrix of the latter is unitary), thus extending previous works that were able to deal only with n=4 and n=5 amplitudes.

On uniform circuit complexity

This paper examines the circuit complexity of feedforward neural networks having sigmoid activation function. The starting point is the complexity class NN k defined in [18]. First two additional complexity classes N N Δ k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315784076/1ea703af-0c66-41e4-8ff3-9e9c700ac445/content/eq1772.tif/> and N N Δ , e k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315784076/1ea703af-0c66-41e4-8ff3-9e9c700ac445/content/eq1773.tif/> having less restrictive conditions (than NN k ) concerning fan-in and accuracy are defined. We then prove several relations among these three classes and well established circuit complexity classes. All the proofs are constructive as being built around binary adders. By relaxing the fan-in condition from logarithmic [18] to (almost) polynomial, the results show that substantially larger classes of sigmoid activation feedforward neural networks can be implemented in polynomial size Boolean circuits. This is done on the expense of a logarithmic factor increase in the number of layers, but having constant fan-in (of 2). The main conclusion is that there are interesting fan-in dependent depth-size tradeoffs when trying to digitally implement sigmoid activation neural networks.

We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in mathsf {Quasi}text {-}mathsf {NP} = mathsf {NTIME}[n^{(log n)^{O(1)}}] and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes mathcal { C}, by showing that mathcal { C} admits non-trivial satisfiability and/or # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial # SAT algorithm for a circuit class {mathcal C}. Say that a symmetric Boolean function f(x1,…,xn) is sparse if it outputs 1 on O(1) values of {sum }_{i} x_{i}. We show that for every sparse f, and for all “typical” mathcal { C}, faster # SAT algorithms for mathcal { C} circuits imply lower bounds against the circuit class f circ mathcal { C}, which may be stronger than mathcal { C} itself. In particular: # SAT algorithms for nk-size mathcal { C}-circuits running in 2n/nk time (for all k) imply NEXP does not have (f circ mathcal { C})-circuits of polynomial size.# SAT algorithms for 2^{n^{{varepsilon }}}-size mathcal { C}-circuits running in 2^{n-n^{{varepsilon }}} time (for some ε > 0) imply Quasi-NP does not have (f circ mathcal { C})-circuits of polynomial size.Applying # SAT algorithms from the literature, one immediate corollary of our results is that Quasi-NP does not have EMAJ ∘ ACC0 ∘ THR circuits of polynomial size, where EMAJ is the “exact majority” function, improving previous lower bounds against ACC0 [Williams JACM’14] and ACC0 ∘THR [Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

Let k be an algebraic number field (of finite degree) and K/k be a (finite) Galois extension with Galois group R. In connection with the class field theory, Weil [101 derived a certain canonical factor set a(o, r) of 9 in the idele-class group (K of K, which is uniquely determined by K/k in the sense of equivalence. For instance, a (mod R') -?f t r(o, A) (Ek k, essentially) (mod NK/k( RK)) is the Artin reciprocity law isomoprhism of 9/9' and Sk/NK/k( SK), where 9' is the commutator subgroup of R. A different approach to this (idele-class) factor set was given in [6]. Let, for each pair o, r, a(o, r) be a representative idele of a(o, r). Then (3a)(p, o, r) = a(o, r)a(po, r)-'a(p, o-r)a(p, oY)is, for each triple p, a, r, a principal idele, i.e. a non-zero element of K. We obtain thus a 3-cochain a(p, a, r) = (6a)(p, a, r) of 9 in the principal idele group PK of K, which is in fact a 3-cocycle. We saw, in [6], that this 3-cycocle a, or more precisely its cohomology class, attached in invariant fashion to the Galois extension K/k, has the order m/m', where m = (K: k) and m' is the least common multiple of the p-degrees of K/k, p running over all primes in k. It was shown further, in a joint paper [4] by Hochschild and the writer, that the cyclic group, of order m/m', generated by this 3-cohomology class is exactly the group of all 3-cohomology classes of 9 in PK with ceiling. This last group is in turn, according to EilenbergMacLane [1], nothing but the group of all Teichmtiller 3-cohomology classes for K/k, belonging to central simple algebras over K to which every element of ft can be extended (as automorphism). Now arises the problem to determine the exact algebra-class (though not unique) which possesses our a, attached to K/k, as its Teichmiiller-class. The present work is devoted to this problem and our solution is given in our Theorem in ?3. If a central simple algebra A over K, to which every automorphism of K/k can be extended, is given in a crossed-product form for instance, we can obtain its Teichmiiller-cocycle in a more or less explicit process involving the HilbertSpeiser theorem. On the other hand, in order to obtain our 3-cocycle a, we repeat the construction of the canonical idele-class factor set a given in [6], which is nothing but the global analogy of the local construction in Hochschild [3]1 and makes use of the idele interpretation of the principal genus theorem of Noether, but with more detailed analysis. Namely, we carry out all the processes in terms of ideles, rather than of idele-classes, preserving thus all principal idele factors which were discarded in [6]. This construction, involving the theorem of everywhere splitting algebras, the Hilbert-Speiser theorem and its idble analogy, leads to a. Though fairly explicit, these constructions of a and the Teichmfiller-

This article is a brief historical report on Teiji Takagi which was prepared at the commencement of ‘Takagi Lectures’ of The Mathematical Society of Japan. The first of its two purposes is to give some informations on the circumstances of education and research of mathematics in Japan surrounding Takagi who could finally established himself as the founder of the Japanese school of modern mathematics. The other is a brief overview on Takagi’s works of mathematics some of which are still attractive to and influential on especially ambitious students of mathematics. The author hopes that careful readers may find some hints for the questions how and why Takagi was able to establish his class field theory. At the end of this article the readers will find an English translation of the preface of his book Algebraic theory of numbers (in Japanese) which is the only thing that he left for us to see his total view over class field theory after the establishment of Artin’s reciprocity law.

Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem

We establish class field theory for three-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.