Narrow Results

This research focuses on the proof of the formula to calculate the number of solutions of the congruence of homogeneous quadratic polynomial with prime ideal modulus of a ring of integers. The study approaches the problem naturally through relatively elementary results including those from number theory and quadratic forms to construct the formula to calculate the number of solutions of the aforementioned congruence.

The purpose of this paper is to present various algebraic views of multisets, and certain connections between the theory of multisets (with multiplicities in the semiring of positive integers) and natural computing, in particular membrane and DNA computing. We introduce a Gödel encoding of multisets, and find some results regarding this encoding together with new connections and interpretations. We also introduce the norm of a multiset and we find some relationships between multiset theory and number theory.

We present the mathematical construction of the physically relevant quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered (the unitarity regime). The subject has several precursors in the mathematical literature. We proceed through an operator-theoretic construction of the self-adjoint extensions of the minimal operator obtained by restricting the free Hamiltonian to wave-functions that vanish in the vicinity of the coincidence hyperplanes: all extensions thus model an interaction precisely supported at the spatial configurations where particles come on top of each other. Among them, we select the physically relevant ones, by implementing in the operator construction the presence of the specific short-scale structure suggested by formal physical arguments that are ubiquitous in the physical literature on zero-range methods. This is done by applying at different stages the self-adjoint extension schemes à la Kreĭn–Višik–Birman and à la von Neumann. We produce a class of canonical models for which we also analyze the structure of the negative bound states. Bosonicity and zero range combined together make such canonical models display the typical Thomas and Efimov spectra, i.e. sequence of energy eigenvalues accumulating to both minus infinity and zero. We also discuss a type of regularization that prevents such spectral instability while retaining an effective short-scale pattern. Besides the operator qualification, we also present the associated energy quadratic forms. We structured our analysis so as to clarify certain steps of the operator-theoretic construction that are notoriously subtle for the correct identification of a domain of self-adjointness.

This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93], where we introduced quadratic forms on a module $V$ over a supertropical semiring $R$ and analyzed the set of bilinear companions of a quadratic form $q:V\to R$ in case the module $V$ is free, with fairly complete results if $R$ is a supersemifield. Given such a companion $b$, we now classify the pairs of vectors in $V$ in terms of $(q,b).$ This amounts to a kind of tropical trigonometry with a sharp distinction between the cases for which a sort of Cauchy–Schwarz (CS) inequality holds or fails. This distinction is governed by the so-called CS-ratio $CS(x,y)$ of a pair of anisotropic vectors $x,y$ in $V$. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93]) of a quadratic form on a free module $X$ over a field in the simplest cases of interest where $rk(X)=2$.

In the last part of the paper, we introduce a suitable equivalence relation on $V\backslash \{0\}$, whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic $x,y\in V$ the CS-ratio $CS(x,y)$ depends only on the rays of $x$ and $y$. We develop essential basics for a kind of convex geometry on the ray-space of $V$, where the CS-ratios play a major role.

In this paper, we construct the solutions of semilinear parabolic PDEs with singular coefficients and establish the link to solutions of backward stochastic differential equations.

We present a set of generators for the symplectic group which is different from the well-known set of transvections, from which the chain equivalence for quadratic forms in characteristic 2 is an immediate result. Based on the chain equivalences for quadratic forms, both in characteristic 2 and not 2, we provide chain equivalences for tensor products of quaternion algebras over fields with no nontrivial 3-fold Pfister forms. The chain equivalence for biquaternion algebras in characteristic 2 is also obtained in this process, without any assumption on the base-field.

We show that a non-hyperbolic quadratic pair on a central simple algebra Brauer equivalent to a quaternion algebra stays non-hyperbolic over some splitting field of the quaternion algebra. This extends a result previously only known for fields of characteristic different from two. Our presentation is free from restrictions on the characteristic of the base field.

We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree $p$ over a field $F$ with $char(F)=p$ is a finite integer $d$ greater than 1 then the symbol length of $p$-algebras of exponent $p$ over $F$ is bounded from above by $\lceil \frac{d-1}{p}\rceil -1$, and show that every two tensor products of symbol algebras of lengths $k$ and $\ell$ with $(k+\ell )p\ge d-1$ can be modified so that they share a common slot. For $p=2$, we obtain an upper bound of $\frac{u(F)}{2}-1$ for the symbol length, which is sharp when $I_q^{3}F=0$.

We deduce an analogue of Quillen–Suslin’s local-global principle for the transvection subgroups of the general quadratic (Bak’s unitary) groups. As an application, we revisit the result of Bak–Petrov–Tang on injective stabilization for the ${\mathrm{\text{K}}}_1$-functor of the general quadratic groups.

This paper expands the theory of quadratic forms on modules over a semiring $R$, introduced in [11–13], especially in the setup of tropical and supertropical algebra. Isometric linear maps induce subordination on quadratic forms, and provide a main tool in our current study. These maps allow lifts and pushdowns of quadratic forms on different modules, preserving basic characteristic properties.

For various types of field extensions $E/F$ and values of $n\in \mathbb{N}$ we consider quadratic forms $\phi$ that lie in the $n$th power $I^{n}E$ of the fundamental ideal of $E$ but are already defined over $F$. We then search for some $\psi \in I^{n}F$ of minimal dimension that maps to $\phi$ when extending the scalars to $E$. This problem can be easily solved completely for finite extensions of odd degree. For quadratic extensions $E/F$, the situation is more involved, but solved for $n\le 3$. For $n>3$, we construct quadratic field extensions $E/F$ and a form $\phi \in I^{n}E$ such that any form $\psi \in I^{n}F$ that maps to $\phi$ when extending the scalars to $E$ has dimension at least $dim\phi +2^{n-2}$.

In this paper, we construct a holomorphic Siegel modular form of weight 2 and level 2, and compute its Fourier coefficients explicitly. Moreover, we prove that this modular form equals the generating function of the representative number $\rho (n,m,r)$ associated with the maximal order in the quaternion algebra ${(-1,-1)}_{\mathbb{Q}}$. As a corollary, we can give a new proof of the famous formula for the sums of three squares. As applications, we give an explicit formula for the numbers of solutions of two systems of Diophantine equations related with Sun’s “1-3-5 conjecture”. Furthermore, we show that “a perfect square” in the integral condition version of Sun’s conjecture can be replaced by “a power of 4”.

This is a series of five lectures around the common subject of the construction of self-adjoint extensions of symmetric operators and its applications to Quantum Physics. We will try to offer a brief account of some recent ideas in the theory of self-adjoint extensions of symmetric operators on Hilbert spaces and their applications to a few specific problems in Quantum Mechanics.

A generalization of Witt's theorem is given. The equivalence of the generalization of Witt's theorem over ℝ and Sylvester's law of nullity is proved.

Let N ≥ 2 be an integer, F a quadratic form in N variables over , and an L-dimensional subspace, 1 ≤ L ≤ N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z,F). This provides an analogue over of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over . We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over . This extends previous results of the author over number fields. All bounds on height are explicit.

We give an elementary proof of the number of representations of an integer by the quaternary quadratic form x² + xy + y² + z² + zt + t².

We study annihilating polynomials and annihilating ideals for elements of Witt rings for groups of exponent 2. With the help of these results and certain calculations involving the Clifford invariant, we are able to give full sets of generators for the annihilating ideal of both the isometry class and the equivalence class of an arbitrary quadratic form over a local field. By applying the Hasse–Minkowski theorem, we can then achieve the same for an arbitrary quadratic form over a global field.

Given a lattice in the plane, we consider zeta-functions encoding the number of well-rounded sublattices of a given index. We are particularly interested in the abscissa of convergence of this function and show that the quality of convergence is related to arithmeticity questions concerning the ambient lattice. In particular, we discover that there are infinitely many similarity classes of well-rounded sublattices in a plane lattice if there is at least one. This generalizes results about the rings of Gaussian and of Eisenstein integers by Fukshansky and his coauthors.

In this study we use some known convolution sums to find the representation number for each of the three octonary quadratic forms , and .

In this study we derive formulae for the number of representations of n by the octonary quadratic forms for any n ∈ ℕ₀.