Construct Examples Research Articles

Positive Hermitian curvature flow on nilpotent and almost-abelian complex Lie groups

We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger-Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian ideal. Finally, we study solitons in the almost-abelian setting. We prove uniqueness and completely classify all left-invariant, almost-abelian solitons, giving a method to construct examples in arbitrary dimensions, many of which admit co-compact lattices.

ПРО ПОРУШЕННЯ ЄДИНОСТІ РОЗВ’ЯЗКУ ЗАДАЧІ ДІРІХЛЕ ДЛЯ СИСТЕМ ДРУГОГО ПОРЯДКУ

The study of the issues of the correct posedness of boundary value problems for differential equations and systems occupies an important place in modern research. When considering correctness, the question of unique solvability of this problem is of paramount importance. In particular, the problem of violation of the uniqueness of the solution of boundary value problems for general differential equations in bounded domains with algebraic boundary is of interest. The property of nontrivial solvability of the homogeneous Dirichlet problem for incorrectly elliptic equations of the second order was first pointed out by A. V. Bitsadze, having constructed an example of an equation with constant complex obtained a condition for the violation of the uniqueness of the solution to the Dirichlet problem in the unit disc for a hyperbolic equation in the case when the slope angles of the characteristics differ in sign. V. P. Burskii, considering the homogeneous Dirichlet problem in the unit disc for second-order equations with constant complex coefficients and a homogeneous non-degenerate symbol, obtained a criterion for nontrivial solvability in the form of π-irrationality of the angle between the characteristics. In this paper, we investigate the question of violation of the uniqueness of the solution of the homogeneous Dirichlet problem for a system of typeless second-order partial differential equations in a model domain – a circle. The original system is written in the form of an equation with commuting matrix coefficients. The permutability condition allows one to obtain a necessary and sufficient condition for the nontrivial solvability of the problem under consideration in the form of equality to zero of the determinant, the elements of which are expressed in terms of the coefficients of the equation. This form of writing the criterion allows one to construct examples of systems for which the kernel of the Dirichlet problem is nontrivial and infinite-dimensional. The study was based on the integral condition for the connection of associated boundary L-traces, as well as a functional scheme, the application of which reduces the expansion of a matrix function in a Fourier series to a standard expansion of each of its elements. A theorem of nontrivial solvability of the homogeneous Dirichlet problem is proved.

Intrinsic density, asymptotic computability, and stochasticity

AbstractThere are many computational problems which are generally “easy” to solve but have certain rare examples which are much more difficult to solve. One approach to studying these problems is to ignore the difficult edge cases. Asymptotic computability is one of the formal tools that uses this approach to study these problems. Asymptotically computable sets can be thought of as almost computable sets, however every set is computationally equivalent to an almost computable set. Intrinsic density was introduced as a way to get around this unsettling fact, and which will be our main focus.Of particular interest for the first half of this dissertation are the intrinsically small sets, the sets of intrinsic density $0$ . While the bulk of the existing work concerning intrinsic density was focused on these sets, there were still many questions left unanswered. The first half of this dissertation answers some of these questions. We proved some useful closure properties for the intrinsically small sets and applied them to prove separations for the intrinsic variants of asymptotic computability. We also completely separated hyperimmunity and intrinsic smallness in the Turing degrees and resolved some open questions regarding the relativization of intrinsic density.For the second half of this dissertation, we turned our attention to the study of intermediate intrinsic density. We developed a calculus using noncomputable coding operations to construct examples of sets with intermediate intrinsic density. For almost all $r\in (0,1)$ , this construction yielded the first known example of a set with intrinsic density r which cannot compute a set random with respect to the r-Bernoulli measure. Motivated by the fact that intrinsic density coincides with the notion of injection stochasticity, we applied these techniques to study the structure of the more well-known notion of MWC-stochasticity.Abstract prepared by Justin Miller.E-mail: jmille74@nd.eduURL: https://curate.nd.edu/show/6t053f4938w

Cylinder curves in finite holonomy flat metrics

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order [Formula: see text], the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when [Formula: see text] is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core curves form an infinite diameter subset of the curve complex. In this paper we focus on the case [Formula: see text] and construct examples illustrating a range of behaviors for the embedded cylinder curves. We prove that if [Formula: see text] and the surface is fully punctured, then the embedded cylinder curves form a finite diameter subset of the curve complex. The same analysis shows that the embedded cylinder curves can only have infinite diameter when the metric has a very specific form. Using this we characterize precisely when the embedded cylinder curves accumulate on a point in the Gromov boundary.

Multiple blowing-up solutions to critical elliptic systems in bounded domains

We construct families of blowing-up solutions to elliptic systems on smooth bounded domains in the Euclidean space, which are variants of the critical Lane-Emden system and analogous to the Brezis-Nirenberg problem. We find a function which governs blowing-up points and rates, observing that it reflects the strong nonlinear characteristic of the system. By using it, we also prove that a single blowing-up solution exists in general domains, and construct examples of contractible domains where multiple blowing-up solutions are allowed to exist. We believe that a variety of new ideas and arguments developed here will help to analyze blowing-up phenomena in related Hamiltonian-type systems.

Fixed Point Of Contractive Mappings Of Integral Type Over An S^{JS}- Metric Space

We obtain sufficient conditions for existence of fixed points of integral type contractive mappings on an S^{JS}- metric spaces. We also study common fixed point and couple fixed point of integral type mappings and construct examples to support our results.

Phase Codes for Mitigating Ambiguities in Range and Velocity

AbstractWe review cubic phase codes for mitigating ambiguities in range and velocity before introducing two specific codes. The two have periodicities of 5 and 7 samples for both the transmitted and the modulation code sequences. The short periods are suitable for generating codes of arbitrary length starting with about 15. We abbreviate the two codes with L5 and L7 and describe generation of the codes starting with kernels (i.e., minimum length sequences that repeat to generate the codes of desired lengths). The L5 modulation code produces 5 spectral replicas of the coded signal and the L7 produces 7. We apply the L7 code to a sinusoid and reveal spectra of the modulated signals from several ambiguous range intervals. Through simulation, we show application to weatherlike signals and construct examples whereby two weather signals and ground clutter are overlaid. Using theory, we define the operating region of the codes in the signal parameter space. The region covers a wide range of overlaid returned powers and spectrum widths; it is obtained from simulations involving the L codes and the SZ(8/64) code. The technique is effective in distinguishing the returns from two trip regions separated by no more than L − 2 ambiguous range intervals and reconstructing the corresponding spectral moments. The L5 and L7 codes protect from trip returns up to the fifth and seventh, making them suitable for short-wavelength (3 and 5 cm) radars as their PRTs must be relatively short to accommodate the expected spread of velocities in storms.

Generalising the Scattered Property of Subspaces

Let V be an r-dimensional \({\mathbb{F}_{{q^n}}}\)-vector space. We call an \({\mathbb{F}_{q}}\)-subspace U of V h-scattered if U meets the h-dimensional \({\mathbb{F}_{{q^n}}}\)-subspaces of V in \({\mathbb{F}_{q}}\)-subspaces of dimension at most h. In 2000 Blokhuis and Lavrauw proved that \({\dim_{\mathbb{F}_{q}}}\) U ≤ rn/2 when U is 1-scattered. Sub-spaces attaining this bound have been investigated intensively because of their relations with projective two-weight codes and strongly regular graphs. MRD-codes with a maximum idealiser have also been linked to rn/2-dimensional 1-scattered subspaces and to n-dimensional (r − 1)-scattered subspaces.In this paper we prove the upper bound rn/(h + 1) for the dimension of h-scattered subspaces, h > 1, and construct examples with this dimension. We study their intersection numbers with hyperplanes, introduce a duality relation among them, and study the equivalence problem of the corresponding linear sets.

SCOTT COMPLEXITY OF COUNTABLE STRUCTURES

AbstractWe define the Scott complexity of a countable structure to be the least complexity of a Scott sentence for that structure. This is a finer notion of complexity than Scott rank: it distinguishes between whether the simplest Scott sentence is $\Sigma _{\alpha }$ , $\Pi _{\alpha }$ , or $\mathrm {d-}\Sigma _{\alpha }$ . We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is $\Sigma _{\lambda + 1}$ for $\lambda $ a limit ordinal. This answers a question left open by A. Miller.We also construct examples of computable structures of high Scott rank with Scott complexities $\Sigma _{\omega _1^{CK}+1}$ and $\mathrm {d-}\Sigma _{\omega _1^{CK}+1}$ . There are three other possible Scott complexities for a computable structure of high Scott rank: $\Pi _{\omega _1^{CK}}$ , $\Pi _{\omega _1^{CK}+1}$ , $\Sigma _{\omega _1^{CK}+1}$ . Examples of these were already known. Our examples are computable structures of Scott rank $\omega _1^{CK}+1$ which, after naming finitely many constants, have Scott rank $\omega _1^{CK}$ . The existence of such structures was an open question.

Concentration phenomena for the fractional Q‐curvature equation in dimension 3 and fractional Poisson formulas

We study the compactness properties of metrics of prescribed fractional $Q$-curvature of order $3$ in $\R^3$. We will use an approach inspired from conformal geometry, seeing a metric on a subset of $\R^3$ as the restriction of a metric on $\R^4_+$ with vanishing fourth-order $Q$-curvature. We will show that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function $\Phi$ in $\R^4_+$), in analogy with a $4$-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.

An asymptotic vanishing theorem for the cohomology of thickenings

Let X be a closed equidimensional local complete intersection subscheme of a smooth projective scheme Y over a field, and let \(X_t\) denote the t-th thickening of X in Y. Fix an ample line bundle \(\mathcal {O}_Y(1)\) on Y. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer c, such that for all integers \(t \geqslant 1\), the cohomology group \(H^k(X_t,\mathcal {O}_{X_t}(j))\) vanishes for \(k < \dim X\) and \(j < -ct\). Note that there are no restrictions on the characteristic of the field, or on the singular locus of X. We also construct examples illustrating that a linear bound is indeed the best possible, and that the constant c is unbounded, even in a fixed dimension.

Backward Shift and Nearly Invariant Subspaces of Fock-type Spaces

Abstract We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces ${\mathcal{F}}_W^p$, whose weight is not necessarily radial. We show that in the spaces ${\mathcal{F}}_W^p$, which contain the polynomials as a dense subspace (in particular, in the radial case), all nontrivial backward shift invariant subspaces are of the form $\mathcal{P}_n$, that is, finite-dimensional subspaces consisting of polynomials of degree at most $n$. In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to zero exponential type), we establish an analogue of de Branges’ ordering theorem. We then construct examples that show that the result fails for general Fock-type spaces of larger growth.

Free partition functions and an averaged holographic duality

We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

On Necessary Conditions of Finite-Valued Random Variable Algebraic Approximation

We consider transformations of random variables on finite sets by algebraic operations. A system of operations is said to be approximation complete if any random variable may be approximated with arbitrary precision by applying the given operations to mutually independent identically distributed random variables whose distributions have no zero components. We establish some necessary conditions for a function system to be approximation complete and construct examples of approximation incomplete systems.

Riemannian Warped Product Submersions

In this paper, we introduce Riemannian warped product submersions and construct examples and give fundamental geometric properties of such submersions. On the other hand, a necessary and sufficient condition for a Riemannian warped product submersion to be totally geodesic, totally umbilic and minimal is given.

On constructing CAP-subgroups in direct products

The cover-avoidance property and its variations have become central in the study of finite groups. Along with this widespread attention comes a need to construct examples and counterexamples. The main goal of this work is to apply the theory of direct products to systematically facilitate such constructions.

Quantitative Estimates on the Singular Sets of Alexandrov Spaces

Let $$X\in \text {Alex}\,^n(-1)$$ be an n-dimensional Alexandrov space with curvature $$\ge -1$$ . Let the r-scale $$(k,\epsilon )$$ -singular set $${\mathcal {S}}^k_{\epsilon ,\,r}(X)$$ be the collection of $$x\in X$$ so that $$B_r(x)$$ is not $$\epsilon r$$ -close to a ball in any splitting space $$\mathbb {R}^{k+1}\times Z$$ . We show that there exists $$C(n,\epsilon )>0$$ and $$\beta (n,\epsilon )>0$$ , independent of the volume, so that for any disjoint collection $$\big \{B_{r_i}(x_i):x_i\in {\mathcal {S}}_{\epsilon ,\,\beta r_i}^k(X)\cap B_1, \,r_i\le 1\big \}$$ , the packing estimate $$\sum r_i^k\le C$$ holds. Consequently, we obtain the Hausdorff measure estimates $${\mathcal {H}}^k({\mathcal {S}}^k_\epsilon (X)\cap B_1)\le C$$ and $${\mathcal {H}}^n\big (B_r ({\mathcal {S}}^k_{\epsilon ,\,r}(X))\cap B_1(p)\big )\le C\,r^{n-k}$$ . This answers an open question in Kapovitch et al. (Metric-measure boundary and geodesic flow on Alexandrov spaces. arXiv:1705.04767 (2017)). We also show that the k-singular set $$\textstyle{\mathcal {S}}^k(X)=\bigcup_{\epsilon>0}\big(\bigcap_{r>0}{\mathcal {S}}^k_{\epsilon ,\,r}\big)$$ is k-rectifiable and construct examples to show that such a structure is sharp. For instance, in the $$k=1$$ case we can build for any closed set $$T\subseteq \mathbb {S}^1$$ and $$\epsilon >0$$ a space $$Y\in \text {Alex}^3(0)$$ with $${\mathcal {S}}^{1}_\epsilon (Y)=\phi (T)$$ , where $$\phi :\mathbb {S}^1\rightarrow Y$$ is a bi-Lipschitz embedding. Taking T to be a Cantor set it gives rise to an example where the singular set is a 1-rectifiable, 1-Cantor set with positive 1-Hausdorff measure.

M-theory and orientifolds

We construct the M-Theory lifts of type IIA orientifolds based on K3-fibred Calabi-Yau threefolds with compatible involutions. Such orientifolds are shown to lift to M-Theory on twisted connected sum G2 manifolds. Beautifully, the two building blocks forming the G2 manifold correspond to the open and closed string sectors. As an application, we show how to use such lifts to explicitly study open string moduli. Finally, we use our analysis to construct examples of G2 manifolds with different inequivalent TCS realizations.

The behavior of $pi$-submaximal subgroups under homomorphisms with $pi$-separable kernels

We explore the extent to which constructing the inductive theory of $\mathfrak{X}$-submaximal subgroups is possible. To this end, we study the behavior of $\pi$-submaximal subgroups under homomorphisms with $\pi$-separable kernels and construct examples where such behavior is irregular.

Curves always lie in the plane spanned by Darboux frame

We study curves whose position vector always lies in the planes spanned by Darboux frame along the curves on the surface. For $$\gamma$$ curves on the surface, we define $$\gamma$$ as the osculatingDarbouxcurve when the position vector of $$\gamma$$ always lies in the tangent plane of the surface and define $$\gamma$$ as the rectifyingDarbouxcurve when the position vector of $$\gamma$$ always lies in the Darboux rectifying plane. In this study, we find the geometric properties of the osculating Darboux curve and rectifying Darboux curve, classify these curves into two cases by using special invariants, and we propose approaches to construct examples of these curves by using normalized osculating Darboux vectors and normalized rectifying Darboux vectors. Moreover, we study the classification of osculating Darboux developables and normal developable surfaces under the condition of the existence of osculating Darboux curve and rectifying Darboux curve as a line of curvature and clad helices.