Schemes of finite expansion and universally closed curves

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant KΘ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H1 and H2 have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) KΘ(H1) and KΘ(H2) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ0 be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ0) for different varieties Θ and also the groups of autoequivalences of Θ0. The problem is to describe these groups for different Θ.

Open problems in finite projective spaces

Neural population coding can represent continuous information by neurons with a series of discrete preferred stimuli, and we find that the bell-shaped tuning curve plays an important role in this mechanism. Inspired by this, we incorporate a bell-shaped tuning curve into the discrete group convolution to achieve continuous group equivariance. Simply, we modulate group convolution kernels by Gauss functions to obtain bell-shaped tuning curves. Benefiting from the modulation, kernels also gain smooth gradients on geometric dimensions (e.g., location dimension and orientation dimension). It allows us to generate group convolution kernels from sparse weights with learnable geometric parameters, which can achieve both competitive performances and parameter efficiencies. Furthermore, we quantitatively prove that discrete group convolutions with proper tuning curves (bigger than 1x sampling step) can achieve continuous equivariance. Experimental results show that 1) our approach achieves very competitive performances on MNIST-rot with at least 75% fewer parameters compared with previous SOTA methods, which is efficient in parameter; 2) Especially with small sample sizes, our approach exhibits more pronounced performance improvements (up to 24%); 3) It also has excellent rotation generalization ability on various datasets such as MNIST, CIFAR, and ImageNet with both plain and ResNet architectures.

This chapter provides algebraic background for directly addressing some simple-sounding yet fundamental questions in algebraic geometry. All the questions relate to the set of simultaneous zeros of finitely many polynomials in n variables over a field.Section 1 concerns existence of zeros. The main theorem is the Nullstellensatz, which in part says that there is always a zero if the finitely many polynomials generate a proper ideal and if the underlying field is algebraically closed.Section 2 introduces the transcendence degree of a field extension. If L/K is a field extension, a subset of L is algebraically independent over K if no nonzero polynomial in finitely many of the members of the subset vanishes. A transcendence basis is a maximal subset of algebraically independent elements; a transcendence basis exists, and its cardinality is independent of the particular basis in question. This cardinality is the transcendence degree of the extension. Then L is algebraic over the subfield generated by a transcendence basis. Briefly any field extension can be obtained by a purely transcendental extension followed by an algebraic extension. The dimension of the set of common zeros of a prime ideal of polynomials over an algebraically closed field is defined to be the transcendence degree of the field of fractions of the quotient of the polynomial ring by the ideal.Section 3 elaborates on the notion of separability of field extensions in characteristic p. Every algebraic extension L/K can be obtained by a separable extension followed by an extension that is purely inseparable in the sense that every element x of L has a power xpe for some integer e≥0 with xpe separable over K.Section 4 introduces the Krull dimension of a commutative ring with identity. This number is one more than the maximum number of ideals occurring in a strictly increasing chain of prime ideals in the ring. For K[X1, . . . , Xn] when K is a field, the Krull dimension in n. If P is a prime ideal in K[X1, . . . , Xn], then the Krull dimension of the integral domain R = K[X1, . . . , Xn]/P matches the transcendence degree over K of the field of fractions of R. Thus Krull dimension extends the notion of dimension that was defined in Section 2.Section 5 concerns nonsingular and singular points of the set of common zeros of a prime ideal of polynomials in n variables over an algebraically closed field. According to Zariski’s Theorem, nonsingularity of a point may be defined in either of two equivalent ways—in terms of the rank of a Jacobian matrix obtained from generators of the ideal, or in terms of the dimension of the quotient of the maximal ideal at the point in question factored by the square of this ideal. The point is nonsingular if the rank of the Jacobian matrix is n minus the dimension of the zero locus, or equivalently if the dimension of the quotient of the maximal ideal by its square equals the dimension of the zero locus. Nonsingular points always exist.Section 6 extends Galois theory to certain infinite field extensions. In the algebraic case inverse limit topologies are imposed on Galois groups, and the generalization of the Fundamental Theorem of Galois Theory to an arbitrary separable normal extension L/K gives a one-one correspondence between the fields F with K⊆F⊆L and the closed subgroups of Gal(L/K).KeywordsGalois GroupField ExtensionMinimal PolynomialInverse LimitBasic AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

<!-- *** Custom HTML *** --> This chapter provides algebraic background for directly addressing some simple-sounding yet fundamental questions in algebraic geometry. All the questions relate to the set of simultaneous zeros of finitely many polynomials in $n$ variables over a field. Section 1 concerns existence of zeros. The main theorem is the Nullstellensatz, which in part says that there is always a zero if the finitely many polynomials generate a proper ideal and if the underlying field is algebraically closed. Section 2 introduces the transcendence degree of a field extension. If $L/K$ is a field extension, a subset of $L$ is algebraically independent over $K$ if no nonzero polynomial in finitely many of the members of the subset vanishes. A transcendence basis is a maximal subset of algebraically independent elements; a transcendence basis exists, and its cardinality is independent of the particular basis in question. This cardinality is the transcendence degree of the extension. Then $L$ is algebraic over the subfield generated by a transcendence basis. Briefly any field extension can be obtained by a purely transcendental extension followed by an algebraic extension. The dimension of the set of common zeros of a prime ideal of polynomials over an algebraically closed field is defined to be the transcendence degree of the field of fractions of the quotient of the polynomial ring by the ideal. Section 3 elaborates on the notion of separability of field extensions in characteristic $p$. Every algebraic extension $L/K$ can be obtained by a separable extension followed by an extension that is purely inseparable in the sense that every element $x$ of $L$ has a power $x^{p^e}$ for some integer $e\geq0$ with $x^{p^e}$ separable over $K$. Section 4 introduces the Krull dimension of a commutative ring with identity. This number is one more than the maximum number of ideals occurring in a strictly increasing chain of prime ideals in the ring. For $K[X_1,\dots,X_n]$ when $K$ is a field, the Krull dimension in $n$. If $P$ is a prime ideal in $K[X_1,\dots,X_n]$, then the Krull dimension of the integral domain $R=K[X_1,\dots,X_n]/P$ matches the transcendence degree over $K$ of the field of fractions of $R$. Thus Krull dimension extends the notion of dimension that was defined in Section 2. Section 5 concerns nonsingular and singular points of the set of common zeros of a prime ideal of polynomials in $n$ variables over an algebraically closed field. According to Zariski's Theorem, nonsingularity of a point may be defined in either of two equivalent ways—in terms of the rank of a Jacobian matrix obtained from generators of the ideal, or in terms of the dimension of the quotient of the maximal ideal at the point in question factored by the square of this ideal. The point is nonsingular if the rank of the Jacobian matrix is $n$ minus the dimension of the zero locus, or equivalently if the dimension of the quotient of the maximal ideal by its square equals the dimension of the zero locus. Nonsingular points always exist. Section 6 extends Galois theory to certain infinite field extensions. In the algebraic case inverse limit topologies are imposed on Galois groups, and the generalization of the Fundamental Theorem of Galois Theory to an arbitrary separable normal extension $L/K$ gives a one-one correspondence between the fields $F$ with $K\subseteq F\subseteq L$ and the closed subgroups of $\mathrm{Gal}(L/K)$.

The notion of an w-local functor is used to formulate and prove a theorem which is claimed to encompass Lefschetz's principle in algebraic geometry. In his foundational work, A. Weil formulates Lefschetz's principle as follows: For a given value of the characteristic p [=zero or a prime], every result involving only a finite number of points and of varieties, which has been proved for some choice of the universal [i.e. an algebraically closed field of characteristic p of infinite transcendence degree over the prime field] remains valid without restriction; there is but one algebraic geometry of characteristic p for each value of p, not one algebraic geometry for each universal domain ([6, p. 306]). The purpose of this paper in applied logic is to give a proof of this statement, a proof which is motivated by the typical example with which Weil follows the above statement. Weil states a theorem which he proves for the universal of complex numbers (using complex analytic methods); this is followed by an argument proving the theorem for any universal of characteristic zero, an argument which involves successive extensions of an isomorphism of finitely-generated subfields of two universal domains. This type of argument is one commonly found in logic and usually called the argument. (See [1] for an expository paper on the use of the method.) The theorem of ?1 of this paper may be regarded as a formalization of this argument to prove a general result which may be said to include Lefschetz's principle as stated above. In fact our theorem is just a special case of a theorem of Feferman [3] and our only contribution is the observation that this result has relevance for algebraic geometry. The significance-theoretical, if not practical-of our general theorem lies in the fact that the back-and-forth argument, which enables one to transfer a result from one universal to another, is done once and for all. Then to apply the general theorem to a particular result of algebraic geometry only requires a check that the particular result falls Received by the editors July 28, 1971. AMS (MOS) subject class flcations (1970). Primary 14A99, 02H15.

Since the late 1940's model theory has found numerous applications to algebra. I would like to indicate some of the points of contact between model theoretic methods and strictly algebraic concerns by means of a few concrete examples and typical applications.§1. The Lefschetz principle. Algebraic geometry has proved to be a fruitful source of model theoretic ideas. What exactly is algebraic geometry? We consider a field K, and let Kn be the set of n-tuples (a1 … an) with coordinates ai in K. Kn is called affine n-space over K. Fix polynomials p1 …, pk in K[x1, …, xn] and definethat is V(p1 …, pk) is the locus of common zeroes of the pi in Kn. We call V(Pi …, Pk) the algebraic variety determined by p1, …, pk. With this terminology we may say:Algebraic geometry is the study of algebraic varieties defined over an arbitrary field K. This definition lacks both rigor and accuracy, and we will indicate below how it may be improved.So far we have placed no restrictions on the base field K. Following Weil [4] it is convenient to start with a so-called “universal domain”; in other words take K to be algebraically closed and of infinite transcendence degree over the prime field. Any particular field can of course be embedded in such a universal domain.

Since the late 1940's model theory has found numerous applications to algebra. I would like to indicate some of the points of contact between model theoretic methods and strictly algebraic concerns by means of a few concrete examples and typical applications.§1. The Lefschetz principle. Algebraic geometry has proved to be a fruitful source of model theoretic ideas. What exactly is algebraic geometry? We consider a field K, and let Kn be the set of n-tuples (a1 … an) with coordinates ai in K. Kn is called affine n-space over K. Fix polynomials p1 …, pk in K[x1, …, xn] and definethat is V(p1 …, pk) is the locus of common zeroes of the pi in Kn. We call V(Pi …, Pk) the algebraic variety determined by p1, …, pk. With this terminology we may say:Algebraic geometry is the study of algebraic varieties defined over an arbitrary field K. This definition lacks both rigor and accuracy, and we will indicate below how it may be improved.So far we have placed no restrictions on the base field K. Following Weil [4] it is convenient to start with a so-called “universal domain”; in other words take K to be algebraically closed and of infinite transcendence degree over the prime field. Any particular field can of course be embedded in such a universal domain.

In this paper, we give a survey of the known results concerning the tensor rank of multiplication in finite extensions of finite fields, enriched with some unpublished recent results, and we analyze these to enhance the qualitative understanding of the research area. In particular, we identify and clarify certain partially proved results and emphasise links with open problems in number theory, algebraic geometry, and coding theory. Bibliography: 92 titles.

We explore the evolution of the probability density function (PDF) for an initially deterministic passive scalar diffusing in the presence of a uni-directional, white-noise Gaussian velocity field. For a spatially Gaussian initial profile we derive an exact spatio-temporal PDF for the scalar field renormalized by its spatial maximum. We use this problem as a test-bed for validating a numerical reconstruction procedure for the PDF via an inverse Laplace transform and orthogonal polynomial expansion. With the full PDF for a single Gaussian initial profile available, the orthogonal polynomial reconstruction procedure is carefully benchmarked, with special attentions to the singularities and the convergence criteria developed from the asymptotic study of the expansion coefficients, to motivate the use of different expansion schemes. Lastly, Monte-Carlo simulations stringently tested by the exact formulas for PDF’s and moments offer complete pictures of the spatio-temporal evolution of the scalar PDF’s for different initial data. Through these analyses, we identify how the random advection smooths the scalar PDF from an initial Dirac mass, to a measure with algebraic singularities at the extrema. Furthermore, the Peclet number is shown to be decisive in establishing the transition in the singularity structure of the PDF, from only one algebraic singularity at unit scalar values (small Peclet), to two algebraic singularities at both unit and zero scalar values (large Peclet).

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza tion of Dirichlet's L-functions with a generalization of class field the ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory.

Laser pointing systems for small targets are mainly confronted with two pointing errors, jitter and boresight, arising due to atmospheric turbulence, mechanical vibrations and errors in optical alignment. Knowledge of these parameters provides information about the quality of the pointing and tracking system. In the past, several techniques have been investigated to estimate these parameters from returned laser signals. These include the key ratio technique, the chi-squared method and the maximum likelihood (ML) estimation technique. These techniques have been studied in the literature for Gaussian irradiance profiles. In particular, the ML estimation technique has been found to provide excellent results. In this paper, we extend the ML estimation technique from Gaussian profiles to near-Gaussian irradiance profiles. Our results show that the modified estimator performs much better than a direct application of the original ML estimator.

The Cohen-Macaulay type of Cohen-Macaulay rings

By following ideas of synthetic real projective geometry rather than classical algebraic geometry, maps in a finite-dimensional desarguesian projective space are used to generate normal curves. We aim at solving the problems of classification, automorphic collineations and generating maps of arbitrary non-degenerate normal curves and degenerate normal curves in desarguesian projective planes (also called degenerate conics). Properties of normal curves are shown, on one hand, by using methods of projective geometry as well as linear algebra and, on the other hand, by applying results on the non-existence of certain types of ordinary and generalized polynomial identities with coefficients in a not necessarily commutative field.

In this paper we study the notion of a D-saturated model, which occupies an intermediate position between the notions of a homogeneous model and a saturated model. Both the homogeneous models and the saturated models play a very important role in model theory. For example, the saturated models are the universal domains of the corresponding theories in the sense that is used in algebraic geometry (one can also notice that the algebraically closed fields of infinite transcendence degree, which are the universal domains in algebraic geometry, are the saturated models of the theory of algebraically closed fields); this means that the saturated models realize all what is necessary for the effective study of the corresponding theory.The D-saturated models proved to be useful in various situations. Naturally the question arises on the conditions under which a model is D-saturated. In this paper we indicate some conditions of such sort for weakly o-minimal models and models of stable theories. Namely, for such models we prove that homogeneity and a certain approximation of Dsaturation imply D-saturation.