搜索结果：Almost

For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ in the form $p_1+p_2+m^n=N$, where $p_1$, $p_2$ $-$ prime numbers, $m$ $-$ natural number satisfying the conditions $$ \left|p_k-μ_kN\right|\le H, \quad k=1,2,\qquad \left|m^n-μ_3N\right|\le H,\qquad H \ge N^{1-\frac1{n(n-1)}} {\mathscr{L}}^{\frac{2^{n+1}}{n-1}+n-1},$$ for $μ_1+μ_2+μ_3=1, \ \ μ_i >0, \mathscr{L} = \ln{N}. $ Keywords: Estermann problem, almost proportional summands, short exponential sum of G. Weyl, small neighborhood of centers of major arcs. Bibliography: 20 titles.

In dimension 4, we extend the correspondence between compact nonsingular toric varieties and regular fans to a correspondence between almost complex torus manifolds and families of multi-fans in a geometric way, where an (almost) complex torus manifold is a $2n$-dimensional compact connected (almost) complex manifold equipped with an effective action of a real $n$-dimensional torus $T^n$ that has fixed points. Let $M$ be a 4-dimensional almost complex torus manifold. To $M$, we associate two equivalent combinatorial objects, a family $Δ$ of multi-fans and a graph $Γ$, which encode the data on the fixed point set. We find a necessary and sufficient condition for each of $Δ$ and $Γ$. Moreover, we provide a minimal model and operations for each of $Δ$ and $Γ$. We introduce operations on a multi-fan and a graph that correspond to blow up and down of a manifold, and show that we can blow up and down $M$ to a minimal manifold $M'$ whose weights at the fixed points are unit vectors in $\mathbb{Z}^2$, $Δ$ to a family of minimal multi-fans that has unit vectors only, and $Γ$ to a minimal graph whose edges all have unit vectors as labels. As an application, if $M$ is complex, $Δ$ is a fan an

The paper studies different variants of almost periodicity notion. We introduce the class of eventually strongly almost periodic sequences where some suffix is strongly almost periodic (=uniformly recurrent). The class of almost periodic sequences includes the class of eventually strongly almost periodic sequences, and we prove this inclusion to be strict. We prove that the class of eventually strongly almost periodic sequences is closed under finite automata mappings and finite transducers. Moreover, an effective form of this result is presented. Finally we consider some algorithmic questions concerning almost periodicity.

Let $G$ be an infinite residually finite group. We show that for every minimal equicontinuous Cantor system $(Z,G)$ with a free orbit, and for every minimal extension $(Y,G)$ of $(Z,G)$, there exist a minimal almost 1-1 extension $(X,G)$ of $(Z,G)$ and a Borel equivariant map $ψ:Y\to X$ that induces an affine bijection $ψ^*$ between $M(Y,G)$ and $M(X,G)$, the spaces of invariant probability measures of $(Y,G)$ and $(X,G)$, respectively. If $Y$ is a Cantor set, then $(Y,G)$ and $(X,G)$ are Borel isomorphic, i.e., $ψ^*$ is also a homeomorphism. As an application, we show that the family of Toeplitz subshifts is a test for amenability for residually finite groups, i.e., a residually finite group $G$ is amenable if and only if every Toeplitz $G$-subshift has invariant probability measures.

A virtual link is said to be almost classical (AC) if it has a homologically trivial representative in some thickened surface $Σ\times [0,1]$, where $Σ$ is a closed orientable surface. AC links provide a useful window for observing the geometric topology of virtual knots. Here we take a different approach and look at AC links through the lens of quantum topology. Two adjustments are needed to the existing theory. First, it is necessary to generalize the definition of AC to include virtual tangles and, in particular, virtual braids. Secondly, to distinguish AC and non-AC tangles, the additional structure of quantum supergroups is required. For each Lie superalgebra $\mathfrak{gl}(m|n)$, we define a pair of $U_q(\mathfrak{gl}(m|n))$ Reshetikhin-Turaev functors $Q^{m|n}$, $\widetilde{Q}^{m|n} \circ Zh$ on framed virtual tangles. Here $Zh$ denotes the Bar-Natan $Zh$ construction. These functors unify the Alexander polynomial (AP) of AC links and the generalized Alexander polynomial (GAP) of all virtual links into a single quantum model: $Q^{1|1}$ recovers the AP of an AC link and for any virtual link $K$, $\widetilde{Q}^{1|1}\circ Zh(K)$ is the 2-variable GAP. However, when $(m,n) e (

We consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. We study three variants of the control problem: Bayesian control, in which we have a prior belief about $a$; bounded agnostic control, in which we have no prior belief about $a$ but we assume that $a$ belongs to a bounded set; and fully agnostic control, in which $a$ is allowed to be an arbitrary real number about which we have no prior belief. In the Bayesian variant, a control strategy is optimal if it minimizes a certain expected cost. In the agnostic variants, a control strategy is optimal if it minimizes a quantity called the worst-case regret. For the Bayesian and bounded agnostic variants above, we produce optimal control strategies. For the fully agnostic variant, we produce almost optimal control strategies, i.e., for any $\varepsilon>0$ we produce a strategy that minimizes the worst-case regret to within a multiplicative factor of $(1+\varepsilon)$.

Let $Φ$ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that $Φ$ is $η$-idempotent, namely, $\|Φ^2-Φ\|_{\mathrm{cb}} \leη$, and construct an associated $\varepsilon$-$C^*$ algebra (of almost-invariant observables) for $\varepsilon=O(η)$. This type of structure has the axioms of a unital $C^*$ algebra but the associativity and other axioms involving the multiplication and the unit hold up to $\varepsilon$. We prove that any finite-dimensional $\varepsilon$-$C^*$ algebra $A$ is $O(\varepsilon)$-isomorphic to some genuine $C^*$ algebra $B$. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When $A$ comes from a finite-dimensional $η$-idempotent UCP map $Φ$, the $O(η)$-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel $Φ^*$ into a decoding channel, producing a state on $B$, and an encoding channel.

Maintaining numerical stability in machine learning models is crucial for their reliability and performance. One approach to maintain stability of a network layer is to integrate the condition number of the weight matrix as a regularizing term into the optimization algorithm. However, due to its discontinuous nature and lack of differentiability the condition number is not suitable for a gradient descent approach. This paper introduces a novel regularizer that is provably differentiable almost everywhere and promotes matrices with low condition numbers. In particular, we derive a formula for the gradient of this regularizer which can be easily implemented and integrated into existing optimization algorithms. We show the advantages of this approach for noisy classification and denoising of MNIST images.

We investigate the geometry of a twisting non-shearing congruence of null geodesics on a conformal manifold of even dimension greater than four and Lorentzian signature. We give a necessary and sufficient condition on the Weyl tensor for the twist to induce an almost Robinson structure, that is, the screen bundle of the congruence is equipped with a bundle complex structure. In this case, the (local) leaf space of the congruence acquires a partially integrable contact almost CR structure of positive definite signature. We give further curvature conditions for the integrability of the almost Robinson structure and the almost CR structure, and for the flatness of the latter. We show that under a mild natural assumption on the Weyl tensor, any metric in the conformal class that is a solution to the Einstein field equations determines an almost CR-Einstein structure on the leaf space of the congruence. These metrics depend on three parameters, and include the Fefferman-Einstein metric and Taub-NUT-(A)dS metric in the integrable case. In the non-integrable case, we obtain new solutions to the Einstein field equations, which, we show, can be constructed from strictly almost Kaehler-Einst

An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth of an almost self-centered graph of order $n;$ (2) the maximum independence number of an almost self-centered graph of order $n$ and radius $r;$ (3) the minimum order of a $k$-regular almost self-centered graph and (4) the maximum size of an almost peripheral graph of order $n;$ (5) which numbers are possible for the maximum degree of an almost peripheral graph of order $n;$ (6) the maximum number of vertices of maximum degree in an almost peripheral graph of order $n$ whose maximum degree is the second largest possible. Whenever the extremal graphs have a neat form, we also describe them.

We study the set of almost quantum correlations and their refinements in the simplest tripartite Bell scenario where each party is allowed to perform two dichotomic measurements. In contrast to its bipartite counterpart, we find that there already exist facet Bell inequalities that witness almost quantum correlations beyond quantum theory in this simplest tripartite Bell scenario. Furthermore, we study the relation between the almost quantum set and the hierarchy of supersets to the quantum set due to Navascués-Pironio-Acín (NPA) [Phys. Rev. Lett. 98, 010401 (2007)]. While the former lies between the first and the third level of the NPA hierarchy, we find that its second level does not contain and is not contained within the almost quantum set. Finally, we investigate the hierarchy of refinements to the almost quantum set due to Moroder et al. [Phys. Rev. Lett. 111, 030501 (2013)], which converges to the set of quantum correlations producible by quantum states having positive partial transposition. This allows us to consider (approximations of) the "biseparable" subsets of the almost-quantum set as well as of the quantum set of correlations, and thereby gain further insights on the

The present paper deals with various aspects of the notion of almost Cohen-Macaulay property, which was introduced and studied by Roberts, Singh and Srinivas. We employ the definition of almost zero modules as defined by a value map, which is different from the version of Gabber-Ramero. We prove that, if the local cohomology modules of an algebra $T$ of certain type over a local Noetherian ring are almost zero, $T$ maps to a big Cohen-Macaulay algebra. Then we study how the almost Cohen-Macaulay property behaves under almost faithfully flat extension. As a consequence, we study the structure of $F$-coherent rings of positive characteristic in terms of almost regularity.

Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. As for circle actions, we show that there exists a (directed labeled) multigraph that encodes weights at the fixed points of $M$. This includes the notion of a GKM graph as a special case that weights at each fixed point are pairwise linearly independent. If in addition $k=n$, i.e., $M$ is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. We show that the Hirzebruch $χ_y$-genus $χ_y(M)=\sum_{i=0}^n a_i(M) \cdot (-y)^i$ of an almost complex torus manifold $M$ satisfies $a_i(M) > 0$ for $0 \leq i \leq n$. In particular, the Todd genus of $M$ is positive and there are at least $n+1$ fixed points. Petrie's conjecture asserts that if a homotopy $\mathbb{CP}^n$ admits a non-trivial circle action, its Pontryagin class agrees with that of $\mathbb{CP}^n$. Petrie proved this conjecture if instead it admits a $T^n$-action. We prove that if a $2n$-dimensional almost complex torus manifold $M$ only shares the Euler number with the complex projective space $\mathbb{CP}^n$, an associated graph agrees with that of a linear $T

For almost contact metric or almost paracontact metric manifolds there is natural notion of $η$-normality. Manifold is called $η$-normal if is normal along kernel distribution of characteristic form. In the paper it is proved that $η$-normal manifolds are in one-one correspondence with Cauchy-Riemann almost contact metric manifolds or para Cauchy-Riemann in case of almost paracontact metric manifolds. There is provided characterization of $η$-normal manifolds in terms of Levi-Civita covariant derivative of structure tensor. It is established existence a Tanaka-like connection on $η$-normal manifold with autoparallel Reeb vector field. In particular case contact metric CR-manifold it is usual Tanaka connection. Similar results are obtained for almost paracontact metric manifolds. For manifold with closed fundamental form we shall state uniqueness of this connection. In the last part is studied bi-Legendrian structure of almost paracontact metric manifold with contact characteristic form. It is established that such manifold is bi-Legendrian flat if and only if is normal. There are characterized semi-flat bi-Legendrian manifolds.

In this paper we define almost gentle algebras. They are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gentle algebra is an admissible cut of a unique Brauer configuration algebra and as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one, is the trivial extension of an almost gentle algebra. We show that to every almost gentle algebra A is associated a hypergraph, and that this hypergraph induces the Brauer configuration of the trivial extension of A. Amongst other things, this gives a combinatorial criterion to decide when two almost gentle algebras have isomorphic trivial extensions.

Almost para-Hermitian manifold it is manifold equipped with almost para-complex structure and compatible pseudo-metric of neutral signature. It is considered a class of immersions of almost para-Hermitian manifolds into almost para-Hermitian manifolds. Such immersions are called slant submanifolds. The concept is an analogue of the idea of slant submanifold in almost Hermitian geometry. There are classified pointwise slant surfaces of four dimensional almost para-Hermitian manifold.

The author is planning if possible classify all three-dimensional $(κ,μ)$-manifolds wether contact metric, almost cosymplectic, para-contact metric, almost para-cosymplectic. Of course classification in contact or almost cosymplectic cases already is provdied. Up to authors knowledge there is no classification for para-contact or almost para-cosymplectic $(κ,μ)$. Conjecture is described by the author in coming paper structures provide classification. The main goal however is to show that these three dimensional manifolds are essentially building blocks of higher-dimensional manifolds. The other possiibilty is to introduce class of manifolds which contain both almost contact metric and almost para-contact metric manifolds as proper subclasses.

Let $A$ be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic $p$. Suppose $Δ$ is an infinite set of positive integers, such that $\left(\frac{m}{p}\right)=1$ for $\forall m\in Δ$. If $A$ does not admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of $A$ has endomorphism ring containing $\mathbb{Z}[x]/(x^2-m)$ for some $m\in Δ$. This implies that there are infinitely many places modulo which $A$ is not simple, generalizing the main result of arXiv:1812.11679 to the non-ordinary case. As an another application, we also generalize the $S$-integrality theorem for elliptic curves over number fields, as proved in arXiv:math/0509485, to the setting of abelian surfaces over global function fields.

The role of an expert in the decision-making process is crucial, as the final recommendation depends on his disposition, clarity of mind, experience, and knowledge of the problem. However, the recommendation also depends on their honesty. But what if the expert is dishonest? Then, the answer on how difficult it is to manipulate in a given case becomes essential. In the presented work, we consider manipulation of a ranking obtained by comparing alternatives in pairs. More specifically, we propose an algorithm for finding an almost optimal way to swap the positions of two selected alternatives. Thanks to this, it is possible to determine how difficult such manipulation is in a given case. Theoretical considerations are illustrated by a practical example.