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Our approach to defining the map f:L→ℝ:𝑓→𝐿ℝf\colon L\to\mathbb{R}italic_f : italic_L → blackboard_R is to glue together certain branches of the leaf space L𝐿Litalic_L. This point of view is outlined in Section 5. More formally, in Section 6 we define an ℝℝ\mathbb{R}blackboard_R-bundle with structure group Homeo+(ℝ)... | Finally, in Section 7 we report on computations showing that manifolds satisfying the hypotheses of 1.1 are abundant in the Hodgson-Weeks census. | We used flipper to drill these manifolds along their pseudo-Anosov singularities and produce ideal triangulations of the resulting many-cusped manifolds. Using SnapPy, we performed surgeries with small coefficients on these manifolds satisfying the constraints of 1.1 and identified the resulting manifolds in the Hodgso... | Building on work of Dunfield and Bell, we were able to find 2598 manifolds in the Hodgson-Weeks census which can be constructed by a surgery satisfying the hypotheses of 1.1. This represents about 44.7% of the 5801 non-L-spaces in the Hodgson-Weeks census [HW94, Dun19]. Dunfield and Bell found monodromies for many of t... | Table 3. The overlap in applicability between our method and previously used methods for proving orderability on the non-L-space rational homology spheres in the Hodgson-Weeks census. | A |
From the proof of Theorem B in [drs22] it is clear that Theorem 2.4 also holds in the case that L𝐿Litalic_L is disconnected with finitely many connected components as long as we assume that every component is non-Legendrian since the heart of the argument is purely local around the image of any connected component of ... | We will need the following C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-close version of Theorem 2.4. | These results follow from the following more general theorem about loose Legendrians which states that we can guarantee the existence of an isotopy of small energy, and even C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-approximate any given isotopy. | We will also need the following statement that allows us to approximate a formal Legendrian by a loose Legendrian. | Proof. We will explain how to adjust the arguments in the proof of Theorem 1.2 in [mur12] to prove this theorem. The C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-close part in the case of a fixed loose chart and the fact that we may choose compactly supported homotopies are consequences of ... | A |
\rightarrow\operatorname{Ho}(\mathcal{M})roman_hocolim start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT : roman_Ho ( caligraphic_M start_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) → roman_Ho ( caligraphic_M ). | ℳℳ\mathcal{M}caligraphic_M is a combinatorial model category. Since in the injective model structures the cofibrations are the objectwise cofibrations, the above proposition follows directly. The universal property of −∧𝕃−superscript𝕃-\wedge^{\mathbb{L}}-- ∧ start_POSTSUPERSCRIPT blackboard_L end_POSTSUPERSCRIPT - ... | Then the above map is, up to isomorphism in Ho(ℳ)normal-Hoℳ\operatorname{Ho}(\mathcal{M})roman_Ho ( caligraphic_M ), the diagonal map | Then ℛℛ\mathcal{R}caligraphic_R preserves the monoidal products up to a natural isomorphism, that is, | Theorem 4.1.1 asserts that the outer triangle above commutes up to isomorphism. This will follow once we show that all the small triangles commute up to isomorphism. | D |
\operatorname{lip}(X,d^{\alpha})italic_d start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ( ⋅ , caligraphic_O ) ∈ roman_Lip ( italic_X , italic_d start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ) ⊆ roman_lip ( italic_X , italic_d start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ) by Proposition 4.4. Hence, ϕt=ϕ−... | We now summarize the structure of this paper. In Section 2, some frequently used notations are recalled for the convenience of the reader. In Section 3, we give a brief review of expanding Thurston maps, visual metrics, orbifolds, universal orbifold covering maps, Lattès maps, and the canonical orbifold metric. We also... | In the discussion below, depending on the conditions we will need, we will sometimes say “Let f𝑓fitalic_f, 𝒞𝒞\mathcal{C}caligraphic_C, d𝑑ditalic_d, ϕitalic-ϕ\phiitalic_ϕ, α𝛼\alphaitalic_α satisfy the Assumptions in Section 5.”, and sometimes say “Let f𝑓fitalic_f and d𝑑ditalic_d satisfy the Assumptions in Section... | We state below the hypotheses under which we will develop our theory in most parts of this paper. We will repeatedly refer to such assumptions in the later sections. We emphasize again that not all assumptions are used in every statement in this paper. | is the support of α𝛼\alphaitalic_α. We will only consider orbifolds with S=S2𝑆superscript𝑆2S=S^{2}italic_S = italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, an oriented 2222-sphere, in this paper. | C |
Because all dominant frequencies of xonsubscript𝑥onx_{\mathrm{on}}italic_x start_POSTSUBSCRIPT roman_on end_POSTSUBSCRIPT are preserved in sν∗subscript𝑠subscript𝜈∗s_{\nu_{\ast}}italic_s start_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT, the accuracy of the final result is 𝒪(|ν∗... | Combining these arguments together, we obtain the following theorem regarding the accuracy and complexity of our algorithm. The formal proof can be found in Appendix B. | To have a rather fair comparison between the algorithms, we deliberately choose the parameters to ensure that the runtimes (Tmax,Ttotal)subscript𝑇subscript𝑇total(T_{\max},T_{\text{total}})( italic_T start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT total end_POSTSUBSCRIPT ) of the algorit... | In this paper, we presented a simple and robust algorithm for QPE using compressed sensing. For the single eigenvalue estimation (i.e., QPE), we rigorously established its Heisenberg-limit scaling in Theorem 2 and numerically demonstrated its performance compared to other state-of-the-art QPE algorithms in Sec. 4. Our ... | There are several aspects of evaluating the performance of an SFT algorithm. Its runtime complexity, sample complexity, and resolution are all important ingredients to consider. Here the runtime complexity refers to how long the algorithm takes on a classical computer, the sample complexity measures the number of time ... | A |
The last expression is linear with respect to |v1|2superscriptsubscript𝑣12|v_{1}|^{2}| italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Therefore, either it is constant, or it is minimized for |v1|=0subscript𝑣10|v_{1}|=0| italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ... | We have seen that, by Theorem 1.1, in the special case when g𝑔gitalic_g is a power-law kernel of the form (1.7) for a suitable choice of the parameters α,β𝛼𝛽\alpha,\,\betaitalic_α , italic_β, the unique minimizing measure is the purely atomic measure μ¯¯𝜇\bar{\mu}over¯ start_ARG italic_μ end_ARG uniformly distribut... | Notice now that the projections of the points xisubscript𝑥𝑖x_{i}italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT with 2≤i≤N2𝑖𝑁2\leq i\leq N2 ≤ italic_i ≤ italic_N on the (N−1)𝑁1(N-1)( italic_N - 1 )-dimensional hyperplane ΠΠ\Piroman_Π are the vertices of the standard regular N𝑁Nitalic_N-gon centered in the... | Let us now consider the sum in (3.5). We can assume that N≥3𝑁3N\geq 3italic_N ≥ 3, since the cases N=1, 2𝑁12N=1,\,2italic_N = 1 , 2 are elementary computations. Arguing similarly as before, we get | Summarizing, up to now in the literature existence of optimal sets was either more or less simple by standard methods, as for energies like (1.8), or it was only obtained in special cases where the optimal sets are actually balls, for energies like (1.1). What we are able to do in this paper is to provide a first argum... | C |
Notation: For the rest of this section, T𝑇Titalic_T denotes a regular tree of degree d≥3𝑑3d\geq 3italic_d ≥ 3 rooted at a vertex v0subscript𝑣0v_{0}italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. | In particular, we introduce the notion of a D𝐷Ditalic_D-mixed subtree quasi-isometry which is a type of quasi-isometry from regular trees to themselves. While a precise definition can be found in Section 3, the main idea behind them is the following; having defined the quasi-isometry for vertices v𝑣vitalic_v at dista... | By Lemma 2.8, which states that all quasi-isometries are at bounded distance from order-preserving quasi-isometries, it suffices to show the moreover part for an order-preserving quasi-isometry. So we assume in the following that g𝑔gitalic_g is order-preserving. | Outline. In Section 2 we introduce the relevant notation and prove some of the technical results about quasi-isometries of trees. In particular, we extend a result of [Nai22] and show that any quasi-isometry is at bounded distance from an order-preserving quasi-isometry. In Section 3 we describe mixed-subtree quasi-iso... | In this section, we describe a way of building quasi-isometries, which we call mixed-subtree quasi-isometries, of regular trees to themselves. We further show that any quasi-isometry is at bounded distance from a mixed-subtree quasi-isometry. The key idea behind mixed-subtree quasi-isometries is that they are quasi-iso... | D |
For CCE manifolds in dimension 4444, there is also some interest in studying specific compactifications; refer to [MR1909634, MR3886176, MR4130465, MR4727591] for an incomplete list. | Our approach is partly inspired by Shen and Wang [MR4308060], though there are several key differences. Shen and Wang use conformal transformations and compare the boundary integral (1.4) of the underlying domain with that of the punctured disk. Subsequently, techniques involving the Schwarzian derivative are applied. ... | The functional λ𝜆\lambdaitalic_λ was first studied by Shen and Wang [MR4308060] in a slightly different setting. Using our notation, Shen and Wang have proven the following. | The author would like to thank Biao Ma for introducing the work of Shen and Wang [MR4308060]. The author would also like to thank Hao Fang, Lihe Wang and Biao Ma for helpful discussions, and Hao Fang for all the guidance on writing this paper. | Theorem 1.3 extends Shen and Wang’s work [MR4308060], in which they compute ΛΛ\operatorname{\Lambda}roman_Λ when M𝑀Mitalic_M is simply connected, and separate this case with multiply-connected domains. Our result further characterizes doubly-connected M𝑀Mitalic_M. Theorem 1.3 shows ΛΛ\operatorname{\Lambda}roman_Λ can... | A |
Ψ⟂:=Ψ−⟨Ψ⟩.assignsuperscriptΨperpendicular-toΨdelimited-⟨⟩Ψ\displaystyle\Psi^{\perp}:=\Psi-\langle\Psi\rangle.roman_Ψ start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT := roman_Ψ - ⟨ roman_Ψ ⟩ . | We consider the solution u𝑢uitalic_u to the following elliptic equation with the Robin boundary condition, | Taking the bracket ⟨⋅⟩delimited-⟨⟩⋅\langle\cdot\rangle⟨ ⋅ ⟩ after multiplying the equation in (1.1) with 1111 and v𝑣vitalic_v leads to the following macroscopic equations, | The model (1.1), with i=1𝑖1i=1italic_i = 1 or 2222, is a kinetic description of the probability distribution of a certain system of interacting particles, submitted to an external force derived from the potential ϕitalic-ϕ\phiitalic_ϕ, at time t𝑡titalic_t located at the position x𝑥xitalic_x in the physical space Ω⊂ℝ... | The mathematical study of approximating linear kinetic equations through macroscopic diffusion equations has been investigated in previous works such as [LK74, BLP79, BSS84, DMG87]. It was noticed in [WG15] that the presence of grazing boundaries may cause a breakdown in the diffusive asymptotics within the L∞superscri... | B |
In addition, pt(x,⋅)subscript𝑝𝑡𝑥⋅p_{t}(x,\cdot)italic_p start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_x , ⋅ ) is the transition function of a strong Markov process whose paths are right continuous and have no discontinuities other than jumps. | The proof will be the same as in the case of [27, Thm. 4.2], and is given here for the convenience of the reader. | The approach, as suggested by the anonymous referee, will be to do this in a much more general setting in the following subsection, and then to specialise that result to the case of the heat operator ϵℋθitalic-ϵsubscriptℋ𝜃\epsilon\mathcal{H}_{\theta}italic_ϵ caligraphic_H start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIP... | and 𝒥𝒥\mathcal{J}caligraphic_J may also be unbounded, a different approach than in the proof of [30, Thm. 3.1]. The approach taken here is similar to that of the proof of [5, Lem. 5.1]. | Brownian motion can be modelled by the heat equation, which describes a diffusion process on a given space. One of its mathematical meanings is to be a tool for extracting information about the space from the diffusion equation. Already the spectrum of the corresponding Laplacian operator reveals something about the sp... | A |
Our technique, which was also used by Blecher and Phillips in Section 4 of [3], will be to show that some bicontractive | Lqsuperscript𝐿𝑞L^{q}italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT-operator algebras. We record precise definitions and | algebras that are not representable on any Lqsuperscript𝐿𝑞L^{q}italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT-space for any q∈[1,∞)𝑞1q\in[1,\infty)italic_q ∈ [ 1 , ∞ ). | and whose multiplier algebra is not an Lqsuperscript𝐿𝑞L^{q}italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT-operator algebra | Given that Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-operator algebras include the large and well studied class of C*-algebras, part of the research in this area focuses on understanding what C*-properties and constructions can and cannot be extended to the Lpsuperscript𝐿𝑝L^{p}... | A |
We end the introduction by giving a brief overview of the structure of the rest of the paper. In Section 2, we review some basic facts about representations over local nonarchimedean field, and several local and global functorial lifts. | The goal of this section is to prove Theorem 1.3, which we restate below. We recall the following setup: | In Section 3 and Section 4, we recall the geometric side and the spectral side of the relative trace formula of [XZ23] respectively. Then the proof of Theorem 1.2 is given in Section 5, while the proof of Theorem 1.3 is given in Section 6. | Theorem 1.2 is proved in Section 5, and is based on a relative trace formula proposed in [XZ23]. We first prove Theorem 1.2 when π𝜋\piitalic_π is supercuspidal, by combining a local-global argument and an involution method similar to that of [Xue21, Section 4]. If π𝜋\piitalic_π is a discrete series representation, th... | By comparing (9) with (7), and truncating the spectral side (7) to a finite sum, we obtain (5), as desired. This finishes the proof. | B |
The sum of the lengths of the two links is constant and shape change is produced by the filament sliding through the hinge. | Although the macroscopic effect is a change in the length of each link, this mechanism can be more properly seen as the limit case of a filament with active curvature, with the opening and closing of the two links being produced by changing the curvature near a point in the filament, adding to the device the possibilit... | While a change in curvature has a trivial interpretation in terms of relative displacement of the material points of the swimmer, the same is no longer true for a length-change in a filament, which can be produced, for example, by a stretching of the filament, or by tip-localized growth, or even by the protrusion of a ... | The points of the inner section which, at a given time, are located inside the outer section are not in contact with the fluid (and do not affect the shape of the swimmer), so they are not accounted for in our description of the body at that time. As in the previous model, therefore, elongation corresponds to a change ... | The sum of the lengths of the two links is constant and shape change is produced by the filament sliding through the hinge. | A |
\mathfrak{C}\}{ italic_w italic_ζ ∣ italic_ζ ∈ fraktur_C } × { italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT 1 italic_ζ ∣ italic_ζ ∈ fraktur_C }. | The trivial pattern corresponds to ℭ2superscriptℭ2\mathfrak{C}^{2}fraktur_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT itself. | Consequently, a set of rectangles of a pattern gives a partition of ℭ2superscriptℭ2\mathfrak{C}^{2}fraktur_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. | We often identify a partition of ℭ2superscriptℭ2\mathfrak{C}^{2}fraktur_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT with the corresponding partition of [0,1]2superscript012[0,1]^{2}[ 0 , 1 ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. | The domain and range pattern of a pair of numbered patterns are defined as the patterns that determines the partition of the domain and range set of ℭ2superscriptℭ2\mathfrak{C}^{2}fraktur_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, respectively. | B |
// Approximate the input tensor network 𝗍𝗇(Us)∪𝗍𝗇(Vs)𝗍𝗇subscript𝑈𝑠𝗍𝗇subscript𝑉𝑠{\mathsf{tn}}(U_{s})\cup{\mathsf{tn}}(V_{s})sansserif_tn ( italic_U start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) ∪ sansserif_tn ( italic_V start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) as a binary tree tensor network 𝓧s... | 2: 𝗍𝗇←←𝗍𝗇absent{\mathsf{tn}}\leftarrowsansserif_tn ← a mapping that maps each vertex set to its approximated tensor network | // Approximate the input tensor network 𝗍𝗇(Us)∪𝗍𝗇(Vs)𝗍𝗇subscript𝑈𝑠𝗍𝗇subscript𝑉𝑠{\mathsf{tn}}(U_{s})\cup{\mathsf{tn}}(V_{s})sansserif_tn ( italic_U start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) ∪ sansserif_tn ( italic_V start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) as a binary tree tensor network 𝓧s... | 13: return the final approximated tensor network, 𝗍𝗇(V)𝗍𝗇𝑉{\mathsf{tn}}(V)sansserif_tn ( italic_V ) | 11: 𝗍𝗇(Us∪Ws)←approx_tensor_network(𝗍𝗇(Us)∪𝗍𝗇(Vs),σ(ℰs),{σ(E′):E′∈ℰs},χ,r,A)←𝗍𝗇subscript𝑈𝑠subscript𝑊𝑠approx_tensor_network𝗍𝗇subscript𝑈𝑠𝗍𝗇subscript𝑉𝑠superscript𝜎subscriptℰ𝑠conditional-setsuperscript𝜎superscript𝐸′superscript𝐸′subscriptℰ𝑠𝜒𝑟𝐴{\mathsf{tn}}(U_{s}\cup W_{s})\leftarrow\text... | C |
Fr𝐒𝟓={reflexive Euclidean frames)}\operatorname{Fr}\mathbf{S5}=\{\text{reflexive Euclidean frames})\}roman_Fr bold_S5 = { reflexive Euclidean frames ) } | Noetherian frames}\}roman_Fr bold_GL bold_.3 = { transitive irreflexive non-branching Noetherian frames } | Noetherian frames}\}roman_Fr bold_Grz bold_.3 = { transitive reflexive non-branching Noetherian frames } | Fr𝐆𝐫𝐳.3={transitive reflexive non-branching Noetherian frames}Fr𝐆𝐫𝐳.3transitive reflexive non-branching Noetherian frames\operatorname{Fr}\mathbf{Grz.3}=\{\text{transitive reflexive non-branching % | Fr𝐆𝐋.3={transitive irreflexive non-branching Noetherian frames}Fr𝐆𝐋.3transitive irreflexive non-branching Noetherian frames\operatorname{Fr}\mathbf{GL.3}=\{\text{transitive irreflexive non-branching % | D |
Additionally, while the analytical solutions of the corresponding cubic symmetry of both the concentrated couple (2.20) and the center of dilatation (2.23) are seemingly more difficult than their isotropic counterpart, they can also be used to numerically find the best approximating cubic counterpart ℂcubicsubscriptℂcu... | Patrizio Neff is grateful for the helpful discussions with Michael A. Slawinski (Memorial University of Newfoundland). P. Neff acknowledges support in the framework of the DFG-Priority Programme 2256 “Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials”, Neff 902/10-1, Project-N... | In Table 1, different sizes of domains are considered and the numerical solution of both of our methods, i.e. considering the norm of the displacement ∥u(r)∥delimited-∥∥𝑢𝑟\lVert u(r)\rVert∥ italic_u ( italic_r ) ∥, or the displacement field u(x1,x2)𝑢subscript𝑥1subscript𝑥2u(x_{1},x_{2})italic_u ( italic_x start_P... | In addition, we specify the corresponding displacement on the outside of the domain to be zero. The validity of the employed ratio of 100:1 between the size of the domain and the diameter of the interior circle stems from and is conformed by extensive numerical testing, cf. Table 1. | Figure 5: Instead of applying the load at the origin as done in the analytical solution in Figure 2, in the numerical approximation we apply the load on a small circle at the center of the domain. It remains to choose the diameter of the circle in relation to the width of the domain. (Left) Schematic drawing of the sta... | A |
Assume ℓ(w)>1ℓ𝑤1\ell(w)>1roman_ℓ ( italic_w ) > 1. Then w=sw′𝑤𝑠superscript𝑤′w=sw^{\prime}italic_w = italic_s italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for some s∈S𝑠𝑆s\in Sitalic_s ∈ italic_S and w′∈Wsuperscript𝑤′𝑊w^{\prime}\in Witalic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W such th... | ρ(si)▶ρ(y)z▶𝜌subscript𝑠𝑖𝜌𝑦𝑧\displaystyle\rho(s_{i})\blacktriangleright\rho(y)zitalic_ρ ( italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ▶ italic_ρ ( italic_y ) italic_z | \blacktriangleright\rho(y)= ( italic_ϕ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ( italic_s , italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) italic_ρ ( italic_s ) italic_ρ ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) ▶ italic_ρ ( italic_y ) | \blacktriangleright\rho(y)\right)= italic_ρ ( italic_s ) ▶ ( italic_ρ ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ▶ italic_ρ ( italic_y ) ) | ρ(w)▶ρ(y)▶𝜌𝑤𝜌𝑦\displaystyle\rho(w)\blacktriangleright\rho(y)italic_ρ ( italic_w ) ▶ italic_ρ ( italic_y ) | D |
;\mathbb{H})divide start_ARG roman_d bold_italic_u end_ARG start_ARG roman_d italic_t end_ARG ∈ roman_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( 0 , italic_T ; blackboard_H ), 𝐟∈W1,1([0,T];ℍ)𝐟superscriptW110𝑇ℍ\boldsymbol{f}\in\mathrm{W}^{1,1}([0,T];\mathbb{H})bold_italic_f ∈ roman_W start_POSTSUPERSCRIPT 1 , 1... | Acknowledgments: With great appreciation, the authors would like to thank the anonymous referee for his or her insightful remarks and ideas, which greatly enhanced the quality of the manuscript. M. T. Mohan would like to thank the Department of Science and Technology (DST), India for Innovation in Science Pursuit for I... | Now, we discuss some of the global solvability results available in the literature for the 3D CBF equations and related models in the whole space as well as in periodic domains. The Cauchy problem corresponding to (1.1) in the whole space (with α=0𝛼0\alpha=0italic_α = 0 and β=r𝛽𝑟\beta=ritalic_β = italic_r) is consid... | is one of the biggest open problems in Mathematics (see [16, 18, 26, 38, 43, 44], etc.). In the recent years, several mathematicians came forward with some modifications of the classical 3D NSE and posed problems on global solvability of such models. The Cauchy problem for the Navier–Stokes equations with damping r|𝒖... | The rest of the paper is organized as follows. In the next section, we discuss the functional setting of the problem described in (1.1). After defining necessary function spaces, we define the linear and nonlinear operators and show that these operators satisfy a monotonicity property for r≥3𝑟3r\geq 3italic_r ≥ 3 (see... | A |
𝒖n+1=g(𝒖n).superscript𝒖𝑛1𝑔superscript𝒖𝑛{\bm{u}}^{n+1}=g({\bm{u}}^{n}).bold_italic_u start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT = italic_g ( bold_italic_u start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) . | where the eigenvectors vj∈ℂsubscript𝑣𝑗ℂv_{j}\in\mathbb{C}italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ blackboard_C denote the dynamic modes as Koopman eigenfunctions, λj∈ℂsubscript𝜆𝑗ℂ\lambda_{j}\in\mathbb{C}italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ blackboard_C denote the Ritz eigenvalu... | The linear infinite dimensional Koopman operator [14] maps any function f:ℳ→ℂ:𝑓→ℳℂf:\mathcal{M}\rightarrow\mathbb{C}italic_f : caligraphic_M → blackboard_C into | where 𝒜𝒜\mathcal{A}caligraphic_A is the Koopman operator. Hence, the snapshots form a Krylov sequence 𝒮1={𝒖1,𝒜𝒖1,𝒜2𝒖1,⋯,𝒜J−1𝒖1}subscript𝒮1superscript𝒖1𝒜superscript𝒖1superscript𝒜2superscript𝒖1⋯superscript𝒜𝐽1superscript𝒖1{\mathcal{S}}_{1}=\{{\bm{u}}^{1},{\mathcal{A}}{\bm{u}}^{1},{\mathcal{A}}^{2}{% | The DMD represents the eigendecomposition [27] of an approximating linear operator 𝒜𝒜\mathcal{A}caligraphic_A corresponding to the Schmidt operator [25], which a is a special case of | B |
The actual integration (up to signs) is just the beta function identity; yielding μk(X1,…,Xk)subscript𝜇𝑘subscript𝑋1…subscript𝑋𝑘\mu_{k}(X_{1},\ldots,X_{k})italic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_X start_POSTSUBSCRIPT italic_k end_POSTS... | Let ϕitalic-ϕ\phiitalic_ϕ be a continuous 𝔄𝔄\mathfrak{A}fraktur_A-valued measure of finite variation on the interval I𝐼Iitalic_I. | Assume that ϕitalic-ϕ\phiitalic_ϕ is a continuous 𝔄𝔄\mathfrak{A}fraktur_A-valued measure of finite variation on the interval I𝐼Iitalic_I. | Let ϕitalic-ϕ\phiitalic_ϕ be a continuous 𝔄𝔄\mathfrak{A}fraktur_A-valued measure of finite variation on the interval I𝐼Iitalic_I. | Let ϕitalic-ϕ\phiitalic_ϕ be a continuous 𝔄𝔄\mathfrak{A}fraktur_A-valued measure of finite variation on the interval I𝐼Iitalic_I. | B |
By the parametric transversality theorem, for almost any s∈Γ𝑠double-struck-Γs\in\mathbb{\Gamma}italic_s ∈ blackboard_Γ , the map f~ssubscript~𝑓𝑠\tilde{f}_{s}over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT will be generic, in that it is either totally real or assumes complex tangents a... | For now, we note that both these linear complex subspaces are necessarily one dimensional as x𝑥xitalic_x is a a one dimensional non-degenerate complex tangent and so the holomorphic tangent space ηxsubscript𝜂𝑥\eta_{x}italic_η start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT is a complex line and by the diffeomorphism ... | will also allow us to extend the map F(θ)𝐹𝜃F(\theta)italic_F ( italic_θ ) accordingly to a local tubular neighborhood of the link, since the obstruction to such a local extension is exactly the value of this same integral of f(θ)𝑓𝜃f(\theta)italic_f ( italic_θ ) about the curve. This corresponds exactly the condit... | We believe that this result demonstrates the full (topological) flexibility of real embeddings into complex Euclidean space in the dimension 3. It may also be interesting to now ask if the same kind of flexibility will extend to the structure of the holomorphic hulls for real 3-dimensional submanifolds of complex space... | Complex tangents to an embedding Mk↪ℂn↪superscript𝑀𝑘superscriptℂ𝑛M^{k}\hookrightarrow\mathbb{C}^{n}italic_M start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ↪ blackboard_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT are points x∈M𝑥𝑀x\in Mitalic_x ∈ italic_M so that the tangent space to M𝑀Mitalic_M at x𝑥... | C |
This also applies to the function B𝐵Bitalic_B, even within a given simulation: once B(n)𝐵𝑛B(n)italic_B ( italic_n ) is known for some n𝑛nitalic_n, computing B(n′)𝐵superscript𝑛′B(n^{\prime})italic_B ( italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) for n′>nsuperscript𝑛′𝑛n^{\prime}>nitalic_n start_POSTSU... | The former solves an open question in [7], and establishes that γ𝛾\gammaitalic_γ can be simulated using rational arithmetic only. In both cases the average numbers of consumed inputs and of required operations are very small. | The average number of series terms is also seen to be very small. In view of (16) and (18), updating the partial sum of the series with a new term and computing the corresponding error bound requires around 20202020 arithmetical operations (the exact number of operations depends on implementation details such as whethe... | This paper describes a general algorithm that solves the above problem when there exists a representation of τ𝜏\tauitalic_τ as a series with rational terms. The algorithm is described in §2, and its basic properties are addressed. The complexity of the algorithm is analysed in §3. Application to specific values, inclu... | The application of Algorithm 1 to γ𝛾\gammaitalic_γ answers an open question posed in [7, section 6], namely devising a “natural” experiment with probability of success γ𝛾\gammaitalic_γ. The procedure presented here is “natural” in the sense that it only requires rational arithmetic and its complexity is very small, b... | D |
The double EPW𝐸𝑃𝑊EPWitalic_E italic_P italic_W sextic associated to a strongly smooth K3𝐾3K3italic_K 3 surface S=ℙ6∩G(2,V5)∩Q𝑆superscriptℙ6𝐺2subscript𝑉5𝑄S=\mathbb{P}^{6}\cap G(2,V_{5})\cap Qitalic_S = blackboard_P start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT ∩ italic_G ( 2 , italic_V start_POSTSUBSCRIPT 5 en... | In this section we want to find precise conditions on a Brill-Noether general, ⟨10⟩delimited-⟨⟩10\langle 10\rangle⟨ 10 ⟩-polarized, K3𝐾3K3italic_K 3 surface such that the associated double EPW𝐸𝑃𝑊EPWitalic_E italic_P italic_W sextic XA(S)subscript𝑋𝐴𝑆X_{A(S)}italic_X start_POSTSUBSCRIPT italic_A ( italic_S ) e... | Let (S,H)∈𝒦10𝑆𝐻subscript𝒦10(S,H)\in\mathcal{K}_{10}( italic_S , italic_H ) ∈ caligraphic_K start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT be a ⟨10⟩delimited-⟨⟩10\langle 10\rangle⟨ 10 ⟩-polarized K3𝐾3K3italic_K 3 surface. | We denote by 𝒦10subscript𝒦10\mathcal{K}_{10}caligraphic_K start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT the moduli space of ⟨10⟩delimited-⟨⟩10\langle 10\rangle⟨ 10 ⟩-polarized K3𝐾3K3italic_K 3 surfaces. | Equivalently, a ⟨10⟩delimited-⟨⟩10\langle 10\rangle⟨ 10 ⟩-polarized K3𝐾3K3italic_K 3 surface is associated to a double EPW𝐸𝑃𝑊EPWitalic_E italic_P italic_W sextic which is a hyper-Kähler manifold if and only if it does not lie in six (explicitly described) divisors in the corresponding moduli space. | D |
However, there does not seem to exist theoretical guarantees for the SCMS algorithm to consistently estimate the full ridge set, and, as discussed below (see Section 2.3 and the Appendix), the SCMS algorithm might miss some parts of the ridge, although the point-wise convergence property of SCMS is studied in Zhang and... | We apply our algorithms to a data set of active and extinct volcanoes in Japan available at https://en.wikipedia.org/wiki/List_of_volcanoes_in_Japan. The locations of these volcanoes exhibit a clear filamentary structure with three major branches sharing an intersection. The results using SCMS and our algorithms are sh... | The remaining part of the paper is organized as follows. In Section 2 we introduce the formal definition of ridges. This is followed by our extraction algorithms, whose performance is illustrated using some numerical studies in ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. The m... | Brief outline of this section: As the algorithms are targeting Ridge(f^)^𝑓(\widehat{f})( over^ start_ARG italic_f end_ARG ), while the theoretical target is Ridge(f),𝑓(f),( italic_f ) , we first control the distance between these two sets (see Theorem 6). Then, in Theorem 8, we consider the continuous version of the ... | This important section can be interpreted as providing population level versions of our main convergence results for the proposed algorithms presented above. Indeed, the algorithms can be interpreted as ‘perturbed versions’ of corresponding population level versions. We will discuss the precise meaning of this in what ... | B |
The closure of the orbit 𝕆𝝀,𝜽subscript𝕆𝝀𝜽{\mathbb{O}}_{{\boldsymbol{\lambda}},{\boldsymbol{\theta}}}blackboard_O start_POSTSUBSCRIPT bold_italic_λ , bold_italic_θ end_POSTSUBSCRIPT supports an irreducible | GL(N−1,𝐎)GL𝑁1𝐎{\mathop{\operatorname{\rm GL}}}(N-1,{\mathbf{O}})roman_GL ( italic_N - 1 , bold_O )-equivariant q𝑞qitalic_q-monodromic perverse sheaf. | GL(N−1,𝐎)GL𝑁1𝐎{\mathop{\operatorname{\rm GL}}}(N-1,{\mathbf{O}})roman_GL ( italic_N - 1 , bold_O )-equivariant q𝑞qitalic_q-monodromic perverse sheaf iff (𝛌,𝛉)𝛌𝛉({\boldsymbol{\lambda}},{\boldsymbol{\theta}})( bold_italic_λ , bold_italic_θ ) satisfies the | GL(N−1,𝐎)GL𝑁1𝐎{\mathop{\operatorname{\rm GL}}}(N-1,{\mathbf{O}})roman_GL ( italic_N - 1 , bold_O )-equivariant q𝑞qitalic_q-monodromic perverse sheaf iff the stabilizer of | and GL(N,𝐎)GL𝑁𝐎{\mathop{\operatorname{\rm GL}}}(N,{\mathbf{O}})roman_GL ( italic_N , bold_O )-equivariant | C |
\operatorname*{diag}\{\psi^{\prime}(\bm{r})\}= roman_diag { italic_ψ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( bold_italic_r ) } - roman_diag { italic_ψ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( bold_italic_r ) } bold_italic_X over^ start_ARG bold_italic_A end_ARG bold_italic_X start_POSTSUPERSCRIPT ⊤ end_POSTS... | derivative with respect to yisubscript𝑦𝑖y_{i}italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in (2.1)). | Both the scalar 𝖽𝖿^^𝖽𝖿\hat{\mathsf{df}}over^ start_ARG sansserif_df end_ARG and the matrix 𝑽∈ℝn×n𝑽superscriptℝ𝑛𝑛\bm{V}\in\mathbb{R}^{n\times n}bold_italic_V ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_n end_POSTSUPERSCRIPT | Since the same matrix 𝑨^^𝑨{\widehat{\bm{A}}}over^ start_ARG bold_italic_A end_ARG appears in both the derivatives with respect | With the matrix 𝑨^∈ℝp×p^𝑨superscriptℝ𝑝𝑝{\widehat{\bm{A}}}\in\mathbb{R}^{p\times p}over^ start_ARG bold_italic_A end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_p × italic_p end_POSTSUPERSCRIPT | C |
We now provide an explicit description of the hierarchically hyperbolic group structure that Eαsubscript𝐸𝛼E_{\alpha}italic_E start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT receives from Theorem 6.10. We start by recalling the defining information of a hierarchically hyperbolic structure (Sections 7.1). We embrace a s... | We now provide an explicit description of the hierarchically hyperbolic group structure that Eαsubscript𝐸𝛼E_{\alpha}italic_E start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT receives from Theorem 6.10. We start by recalling the defining information of a hierarchically hyperbolic structure (Sections 7.1). We embrace a s... | Next we describe the hierarchically hyperbolic structure that Theorem 2.18 imparts on an abstract combinatorial HHS (Section 7.2). | This implies that B(Wα)𝐵subscript𝑊𝛼B(W_{\alpha})italic_B ( italic_W start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) is connected and a hierarchically hyperbolic space. We now verify the additional requirements from Theorem 2.18 for Eαsubscript𝐸𝛼E_{\alpha}italic_E start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT to... | When (X,W)𝑋𝑊(X,W)( italic_X , italic_W ) is a combinatorial HHS, the space W𝑊Witalic_W is a hierarchically hyperbolic space. Further, a group G𝐺Gitalic_G will be hierarchically hyperbolic if it acts as described below on both X𝑋Xitalic_X and W𝑊Witalic_W. In Section 7, we provide a brief summary of the salient par... | B |
By Lemma 1.4 a node cannot have two parents. Therefore, each connected component of the Hasse diagram is a tree with root at the top. | Consider the exceptional sequence for A7subscript𝐴7A_{7}italic_A start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT: | Let (E1,⋯,En)subscript𝐸1⋯subscript𝐸𝑛(E_{1},\cdots,E_{n})( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) be an exceptional sequence with sequence of tops (a1,⋯,an)subscript𝑎1⋯subscript𝑎𝑛(a_{1},\cdots,a_{n})( italic_a start_POSTSUBSCRIPT 1 end_POSTS... | We consider the Hasse diagram of an exceptional sequence and derive some of its properties. For example, we show that it is a rooted forest. | Figure 1. By Theorem A1, this figure indicates the rooted labeled forest corresponding to the complete exceptional sequence for the quiver A7subscript𝐴7A_{7}italic_A start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT: | A |
In its minimal genus thickened surface, the knot must have all regions simply connected, or otherwise we could find a reducing annulus. Given a connected alternating projection on an orientable surface with all regions topological disks, it can always be checkerboard colored. In particular, as in Figure 22, each region... | In its minimal genus thickened surface, the knot must have all regions simply connected, or otherwise we could find a reducing annulus. Given a connected alternating projection on an orientable surface with all regions topological disks, it can always be checkerboard colored. In particular, as in Figure 22, each region... | It is impossible for β𝛽\betaitalic_β to enclose only a single puncture from K𝐾Kitalic_K; if a strand of the knot enters the region inside of D𝐷Ditalic_D it must also leave that region. | But if there were a singular curve, it would intersect only one edge, meaning a region would be adjacent to itself, contradicting the fact adjacent regions must have distinct colors. | If there exists a projection P𝑃Pitalic_P of a virtual knot K𝐾Kitalic_K onto a projection surface S𝑆Sitalic_S that intersects a nontrivial curve exactly once, that is a singular curve. But in general, given a projection, it may be difficult to determine if a singular curve exists. Figure 3 gives an example of a singu... | C |
(a) Average convergence time to almost-consensus in a with noise parameter p=1/8𝑝18p=1/8italic_p = 1 / 8. | (b) Average convergence time to almost-consensus in a configuration graph with degree d=5𝑑5d=5italic_d = 5 with noise parameter p=1/4𝑝14p=1/4italic_p = 1 / 4. | (b) Average convergence time to almost-consensus in a with noise parameter p=1/4𝑝14p=1/4italic_p = 1 / 4. | (a) Average convergence time to almost-consensus in a configuration graph with degree d=5𝑑5d=5italic_d = 5 with noise parameter p=1/8𝑝18p=1/8italic_p = 1 / 8. | (a) Average convergence time to almost-consensus in a with noise parameter p=1/8𝑝18p=1/8italic_p = 1 / 8. | B |
While the intersection numbers above are readily computable for many pairs (g,n)𝑔𝑛(g,n)( italic_g , italic_n ) using the software admcycles [DSv20], it proved difficult to guess a formula generalizing the one presented in [BSSZ15]. Alternatively one could try to understand how to naturally account for the twisting pa... | The aim of the present work is to generalize the results of [BSSZ15] and compute integrals of the top power of a single ψ𝜓\psiitalic_ψ-class on strata of k𝑘kitalic_k-differentials and on twisted double ramification cycles. The results we present allow both to simplify the computations appearing in [Sau20] about volum... | In Table 1 we present the orbifold Euler characteristic of the even and odd spin components of minimal strata of abelian differentials in low genera. | A second direction of study is the asymptotic behaviour of the orbifold Euler characteristic of minimal strata of differentials. | In Section 6 below we present a second application of this formula. The first author with M. Möller and J. Zachhuber, provided an expression for the orbifold Euler characteristic of strata of abelian differentials in terms of intersection numbers on these strata (see [CMZ20a]). We use this result together with Theorem ... | C |
Our exercise directly extends the exercise of [Athey et al., 2020a]. Whereas those authors study the long term effect of a binary action (“small” versus “large” classes), we study the long term effect of a continuous action (various class sizes). Figure 1 shows that the randomized action D𝐷Ditalic_D takes values in 𝒟... | As in previous work, we consider the third grade test score to be the short term reward S𝑆Sitalic_S, and a subsequent test score to be the long term reward Y𝑌Yitalic_Y. By choosing different grades as different long term rewards, we evaluate how our methods perform over different horizons. Our variable definitions ar... | To demonstrate that our proposed kernel methods are practical for empirical research, we evaluate their ability to recover long term dose response curves. Using short term experimental data and long term observational data, our methods measure similar long term effects as an oracle method that has access to long term e... | The difficulty in estimating long term dose response curves is the complex nonlinearity and heterogeneity in the link between short term response curve 𝔼{S(d)}𝔼superscript𝑆𝑑\mathbb{E}\{S^{(d)}\}blackboard_E { italic_S start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT } and long term response curve 𝔼{Y(d)}𝔼... | The oracle, visualized in red, is estimated from long term experimental data, i.e. joint observations of the randomized action D𝐷Ditalic_D and long term reward Y𝑌Yitalic_Y in Project STAR. Our goal is to recover similar estimates without access to long term experimental data. Figure 4 shows that the oracle curve is t... | A |
C′⋅D′=∑iaiu(e)(Di)<0.⋅superscript𝐶′superscript𝐷′subscript𝑖subscript𝑎𝑖u𝑒subscript𝐷𝑖0C^{\prime}\cdot D^{\prime}=\sum_{i}a_{i}\textbf{u}(e)(D_{i})<0.italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⋅ italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ital... | C′⋅D′=∑iaiu(e)(Di)<0.⋅superscript𝐶′superscript𝐷′subscript𝑖subscript𝑎𝑖u𝑒subscript𝐷𝑖0C^{\prime}\cdot D^{\prime}=\sum_{i}a_{i}\textbf{u}(e)(D_{i})<0.italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⋅ italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ital... | For t∈τ′𝑡superscript𝜏′t\in\tau^{\prime}italic_t ∈ italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and v′∈V(S′)superscript𝑣′𝑉superscript𝑆′v^{\prime}\in V(S^{\prime})italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_V ( italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), by Corollary 4.8, ht(v... | This contradicts the assumption that D′superscript𝐷′D^{\prime}italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is nef. | Let (X,D)𝑋𝐷(X,D)( italic_X , italic_D ) be an snc log Calabi-Yau with X∖D𝑋𝐷X\setminus Ditalic_X ∖ italic_D a connected smooth affine log Calabi-Yau with Zariski dense torus, with D𝐷Ditalic_D the support of a nef divisor, and consider a birational morphism (X′,D′)→(X,D)→superscript𝑋′superscript𝐷′𝑋𝐷(X^{\prime},D... | C |
We define ℓpsuperscriptℓ𝑝\ell^{p}roman_ℓ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-homology and ℓpsuperscriptℓ𝑝\ell^{p}roman_ℓ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-cohomology as presented in [Bou16b]. | [Bou16b, Section 1.4] Let X𝑋Xitalic_X be a simply connected simplicial complex of bounded geometry. Then for all p>1𝑝1p>1italic_p > 1 we have canonical isomorphisms: | Let X𝑋Xitalic_X be the simplicial realization of a non-spherical regular building 𝒞𝒞\mathcal{C}caligraphic_C of type (W,S)𝑊𝑆(W,S)( italic_W , italic_S ) and finite thickness. The space X𝑋Xitalic_X does not enjoy good topological properties as usually it is not locally finite (see Example 2.9). The following const... | First, let X𝑋Xitalic_X be a simplicial complex equipped with a metric so that it becomes a length space. We say that such a complex X𝑋Xitalic_X has bounded geometry if: | In this subsection let X𝑋Xitalic_X be a contractible, Gromov-hyperbolic simplicial complex with bounded geometry, let ∂X𝑋\partial X∂ italic_X be its Gromov boundary and (⋅|⋅)o(\cdot|\cdot)_{o}( ⋅ | ⋅ ) start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT be the Gromov product on X𝑋Xitalic_X with basepoint o∈X𝑜𝑋o\in Xita... | C |
πesubscript𝜋𝑒\pi_{e}italic_π start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT for the canonical projection from S∗superscript𝑆∗S^{\ast}italic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT onto S+∗superscriptsubscript𝑆∗S_{+}^{\ast}italic_S start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSC... | normalisation mod e𝑒eitalic_e for a monoid M𝑀Mitalic_M, then N¯¯𝑁\overline{N}over¯ start_ARG italic_N end_ARG is | If (S,N)𝑆𝑁\left(S,N\right)( italic_S , italic_N ) is a normalisation mod e𝑒eitalic_e for a monoid M𝑀Mitalic_M, | A normalisation for a monoid M𝑀Mitalic_M is a normalisation (S+,N)subscript𝑆𝑁\left(S_{+},N\right)( italic_S start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_N ) | Then a normalisation (S,N)𝑆𝑁\left(S,N\right)( italic_S , italic_N ) mod e𝑒eitalic_e for M𝑀Mitalic_M is provided | B |
As mentioned in Section 1.3, the results developed in this section are conceptually close to the method of probability metrics (see e.g. Rachev and Römisch (2002)). We discuss in detail the relation to this work in Remark 2. | We note that different classes of functions may induce the same metric. Furthermore, many common metrics (that we define below) such as the Wasserstein distance or the Kolmogorov distance are in fact integral probability metrics. | Focal metrics. The definition above is general, and the framework can be applied to a variety of metrics. In this work, we also aim at developing an understanding of the effect of heterogeneity under various metrics and | Müller (1997) characterizes the maximal generator for many common integral probability metrics. In particular, they show the following. | We focus on a broad class of metrics often referred to as probability metrics with a ζ𝜁\zetaitalic_ζ-structure (Zolotarev, 1984) or integral probability metrics (Müller, 1997; Sriperumbudur et al., 2010). In this work we will use the latter denomination. These metrics are defined as follows. | D |
In the language of Section 6 we may say that the lifted vector fields are “horizontal” and that the horizontal distribution is integrable. | We will use the same notation to denote the corresponding normal vector fields along ΣΣ\Sigmaroman_Σ. | Let ΣΣ\Sigmaroman_Σ be the fibre above b∈B𝑏𝐵b\in Bitalic_b ∈ italic_B. It is natural to view B𝐵Bitalic_B as a submanifold in the moduli space ℳℳ\mathcal{M}caligraphic_M defined by ΣΣ\Sigmaroman_Σ. | In this context there are two natural ways to use e∈𝔰𝑒𝔰e\in\mathfrak{s}italic_e ∈ fraktur_s to deform a coassociative fibre ΣΣ\Sigmaroman_Σ: | In Section 4 we show how this can be done in the context of G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT manifolds satisfying a natural “quadratic condition”, | C |
The problem structure is exploited to decompose the TOP into two smaller, identical and independent problems, i.e., assignment of trolleys and assignment of stackers, by pre-computing the dependency between them. So, a single and smaller MILP model is sufficient to solve both the problems and, hence, to solve the TOP (... | Due to a long manual setup time, the company lacks the flexibility to introduce new PCBs in the build process as well as in responding to unprecedented situations, like COVID-19, for which the company may have to reschedule the build process according to the changed demands. Moreover, the manual setup also leads to a s... | An industrial case study of a company is presented that currently uses a manual setup for the TOP and suffers from long setup times and lack of flexibility to introduce new products and to respond to unprecedented situations, like COVID-19. The proposed model helps the company to automate the TOP and get rid of issues ... | We present a case study of an aerospace company which has an assembly shop to meet its PCB needs across all programmes and operates in a low-volume and high-mix setting. Due to the complex nature of the PCB planning problem and due to the customised needs of the company, all the setups are performed manually by experie... | We compare the results of the proposed model to the current practice, i.e., the manual process used by the company to solve the TOP. The current practice suffers mainly from two issues. First and the most important issue is the long time of eight weeks to prepare the trolley loading setup which causes the assembly shop... | B |
(f25(2),f35(2),g30(2)/g5(2))superscriptsubscript𝑓252superscriptsubscript𝑓352superscriptsubscript𝑔302superscriptsubscript𝑔52(f_{25}^{(2)},f_{35}^{(2)},g_{30}^{(2)}/g_{5}^{(2)})( italic_f start_POSTSUBSCRIPT 25 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT 35 end_POS... | =(275807,1855077841,551532),absent2758071855077841551532=(275807,1855077841,551532),= ( 275807 , 1855077841 , 551532 ) , | For which integers (or rational numbers) a𝑎aitalic_a and b𝑏bitalic_b is the slope of one of the angle bisectors between two straight lines with slopes a𝑎aitalic_a and b𝑏bitalic_b an integer (or a rational number)? | =(275807,54608393,548842),absent27580754608393548842=(275807,54608393,548842),= ( 275807 , 54608393 , 548842 ) , | =(1855077841,12477253282759,3709604150),absent1855077841124772532827593709604150=(1855077841,12477253282759,3709604150),= ( 1855077841 , 12477253282759 , 3709604150 ) , | A |
103⋅(−0.49401,0.99900)⋅superscript1030.494010.9990010^{3}\cdot(-0.49401,0.99900)10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ ( - 0.49401 , 0.99900 ) | 2.50109⋅103⋅2.50109superscript1032.50109\cdot 10^{3}2.50109 ⋅ 10 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT | 9.71940⋅10−5⋅9.71940superscript1059.71940\cdot 10^{-5}9.71940 ⋅ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT | 9.85431⋅10−5⋅9.85431superscript1059.85431\cdot 10^{-5}9.85431 ⋅ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT | 4.28624⋅10−5⋅4.28624superscript1054.28624\cdot 10^{-5}4.28624 ⋅ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT | A |
Let X→B→𝑋𝐵X\rightarrow Bitalic_X → italic_B be a log smooth morphism of fine and saturated log schemes, locally of finite type with X𝑋Xitalic_X locally noetherian over an algebraically closed field 𝕜𝕜\mathds{k}blackboard_k of characteristic 00. | Families of tropical maps in the context of log Gromov-Witten theory typically arise as follows: Suppose we have a punctured log map (C∘/W,f)superscript𝐶𝑊𝑓(C^{\circ}/W,f)( italic_C start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT / italic_W , italic_f ) with target an fs log scheme X𝑋Xitalic_X, or its Artin fan 𝒜Xsubsc... | We let γ𝛾\gammaitalic_γ be a tropical lift of the underlying tropical type τ𝜏\tauitalic_τ. For any fixed decoration 𝜸𝜸\boldsymbol{\gamma}bold_italic_γ of γ𝛾\gammaitalic_γ by curve classes in NE(X~)𝑁𝐸~𝑋NE(\tilde{X})italic_N italic_E ( over~ start_ARG italic_X end_ARG ), by [ACGS20b] Theorem 3.123.123.123.12, ℳ... | After recalling the construction of 𝔐γ→τsubscript𝔐→𝛾𝜏\mathfrak{M}_{\gamma\rightarrow\tau}fraktur_M start_POSTSUBSCRIPT italic_γ → italic_τ end_POSTSUBSCRIPT at the start of Section 4444, 𝔐γ→τsubscript𝔐→𝛾𝜏\mathfrak{M}_{\gamma\rightarrow\tau}fraktur_M start_POSTSUBSCRIPT italic_γ → italic_τ end_POSTSUBSCRIPT is a... | In [ACGS20b] Definition 2.12.12.12.1, the authors introduce the following notion of a puncturing of a log structure: | D |
In a recent breakthrough, Campos, Griffiths, Morris and Sahasrabudhe [4], reduced the the long standing upper bound of 4ksuperscript4𝑘4^{k}4 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT to (4−ε)ksuperscript4𝜀𝑘(4-\varepsilon)^{k}( 4 - italic_ε ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT. This is the fi... | We prove Theorem 1.1 in Section 2 and Theorem 1.4 in Section 3. We conclude with some further discussion in Section 4. | In their paper, Fishburn and Graham [10] introduced another natural generalisation for monotone sequences and the Erdős-Szekeres theorem, which they called a lex-monotone array. | By Ramsey’s Theorem, for each 𝐯∈T′𝐯superscript𝑇′{\mathbf{v}}\in T^{\prime}bold_v ∈ italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, there is a monochromatic copy of Kksubscript𝐾𝑘K_{k}italic_K start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT contained in | In this paper, we prove a “multidimensional” generalisation of Ramsey’s Theorem for cartesian products of graphs. | D |
N}}(|\varphi\rangle+\theta|\eta\rangle)| italic_φ ⟩ ↦ | over~ start_ARG italic_φ end_ARG ⟩ := divide start_ARG 1 end_ARG start_ARG caligraphic_N end_ARG ( | italic_φ ⟩ + italic_θ | italic_η ⟩ ) | shift the state |φ⟩ket𝜑\ket{\varphi}| start_ARG italic_φ end_ARG ⟩ in the direction of |η⟩ket𝜂\ket{\eta}| start_ARG italic_η end_ARG ⟩ by | N}}(|\varphi\rangle+\theta|\eta\rangle)| italic_φ ⟩ ↦ | over~ start_ARG italic_φ end_ARG ⟩ := divide start_ARG 1 end_ARG start_ARG caligraphic_N end_ARG ( | italic_φ ⟩ + italic_θ | italic_η ⟩ ) | |φ~⟩ket~𝜑|\tilde{\varphi}\rangle| over~ start_ARG italic_φ end_ARG ⟩. This process is iterated until the geometric | for |φ~⟩ket~𝜑|\tilde{\varphi}\rangle| over~ start_ARG italic_φ end_ARG ⟩ than for |φ⟩ket𝜑|\varphi\rangle| italic_φ ⟩. Indeed | C |
Feinstein and the author introduced a general method for constructing essential uniform algebras. Using this method they constructed an essential, natural, regular uniform algebra on the closed unit disc D¯¯𝐷\overline{D}over¯ start_ARG italic_D end_ARG [12, Theorem 1.2]. Repeating the proof of [12, Theorem 1.2] with t... | There exists an essential, strongly regular uniform algebra on the closed unit disc D¯¯𝐷\overline{D}over¯ start_ARG italic_D end_ARG. | Feinstein and the author introduced a general method for constructing essential uniform algebras. Using this method they constructed an essential, natural, regular uniform algebra on the closed unit disc D¯¯𝐷\overline{D}over¯ start_ARG italic_D end_ARG [12, Theorem 1.2]. Repeating the proof of [12, Theorem 1.2] with t... | There exists a compact set K𝐾Kitalic_K in the complex plane such that R(K)𝑅𝐾R(K)italic_R ( italic_K ) is a nontrivial strongly regular uniform algebra. | The uniform algebra A𝐴Aitalic_A is strongly regular if A𝐴Aitalic_A is strongly regular at every point of X𝑋Xitalic_X. | A |
As discussed in section 2, both CDT and RCDT have been developed for distributions, respectively, in 1D and 2D. One of the aims of this work is to extend these procedures to generic images to be applied for general model reduction purposes. To this aim, we pre-process all the data in this and the next section, by norma... | For the case of distributions that are non-strictly positive and discontinuous in a finite number of points, the CDF is piecewise invertible and differentiable and the above definitions are valid in a piecewise manner. | We can further extend the CDT for arbitrary positive functions by applying a normalisation before applying the CDT and de-normalise the result after the inversion, and for non-positive functions by performing the CDT transform separately on the positive and negative parts. | Most of the examples presented here are strictly positive fields. However, when the initial image contains negative values, the images are split into positive and negative parts, and the transformation and subsequent steps (POD and interpolation) are applied on each part separately. The final image after inverse transf... | We use a signed variant of RCDT, applying the RCDT-POD to the positive and negative parts of the inputted CFD data separately and later re-combined. See [35] for further details on the transformation of signed measures in RCDT space. This results in a double number of POD modes being used in interpolation and reconstru... | C |
In fact, it has already been noted by the authors of [8] that their results apply not only to locally cyclic graphs of minimum degree ≥7absent7\geq 7≥ 7, but equally to general graphs of local girth ≥7absent7\geq 7≥ 7. | Can the results for locally cyclic graphs of minimum degree δ≥6𝛿6\delta\geq 6italic_δ ≥ 6 be generalized to graphs of local girth ≥6absent6\geq 6≥ 6? | By joining 4.3, 4.5, and Theorem A, we conclude the characterisation of clique convergent locally cyclic graphs with minimum degree δ≥6𝛿6\delta\geq 6italic_δ ≥ 6. | In fact, it has already been noted by the authors of [8] that their results apply not only to locally cyclic graphs of minimum degree ≥7absent7\geq 7≥ 7, but equally to general graphs of local girth ≥7absent7\geq 7≥ 7. | we conclude the characterisation of clique convergent triangularly simply connected locally cyclic graphs of minimum degree δ≥6𝛿6\delta\geq 6italic_δ ≥ 6. | A |
The key idea of this generalization is to select columns from two matrices that share the same column indices in order to extract the features of one dataset that are most pertinent to the other. Here, the common factor matrix produced by the GSVD of the two matrices is used to sample the columns and rows using the Dis... | We have recently generalized the DEIM method to the tensor case based on the t-product [20] that we call tensor DEIM (TDEIM). It has been experimentally shown that the TDEIM achieved the best sampling accuracy compared to the existing sampling techniques, such as top tubal leverage score sampling and uniform sampling. | The key idea of this generalization is to select columns from two matrices that share the same column indices in order to extract the features of one dataset that are most pertinent to the other. Here, the common factor matrix produced by the GSVD of the two matrices is used to sample the columns and rows using the Dis... | A deterministic or random selection of the columns and rows is possible, with the option of obtaining additive or relative approximation errors. It is generally known that the columns or rows with the highest volume might provide virtually optimal solutions in a deterministic scenario [29]. The DEIM is another determin... | Motivated by the works [2, 3], we extend the tensor CUR (TCUR) approximation [18] to a pair and a triplet of tensors based on the t-product and we refer to as the generalized TCUR (GTCUR). As real-world datasets often have multidimensional structure, it is more favorable to generalize the ideas proposed in [2, 19] to t... | A |
The bisector of one of the angles and that of the supplementary angle are perpendicular to each other. | In the case when they are not parallel to the coordinate axes, if one of the slopes is rational, then so is the other, since the product of the slopes is −1.1-1.- 1 . | The bisector of one of the angles and that of the supplementary angle are perpendicular to each other. | For which rational numbers a𝑎aitalic_a and b𝑏bitalic_b are the slopes of the angle bisectors between two straight lines with slopes a𝑎aitalic_a and b𝑏bitalic_b rational? | Since the ideal class group ClK𝐶subscript𝑙𝐾Cl_{K}italic_C italic_l start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT of K𝐾Kitalic_K is finite, and so is the quotient monoid I/∼I/\!\simitalic_I / ∼ which can be regarded as its submonoid. | A |
An important technical and computational tool for Hilbert C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-modules is the concept of a frame. | This is as close as one can get to an orthonormal basis in a C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-module, and it serves similar purposes. | In particular, X(EI)𝑋subscript𝐸𝐼X(E_{I})italic_X ( italic_E start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ) is not isomorphic to X(F)≅X(E)𝑋𝐹𝑋𝐸X(F)\cong X(E)italic_X ( italic_F ) ≅ italic_X ( italic_E ) as C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-correspondences. | This strong shift equivalence is reflected in the associated Cuntz–Pimsner C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-algebras as gauge-equivariant Morita equivalence. | This paper studies noncommutative dynamical systems—defined as C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-correspondences over not necessarily commutative C∗superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT-algebras— | A |
By taking the free product with M2(ℂ)subscript𝑀2ℂM_{2}(\mathbb{C})italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( blackboard_C ) once we have L(𝔽m+34)≡L(𝔽n+34)𝐿subscript𝔽𝑚34𝐿subscript𝔽𝑛34L(\mathbb{F}_{m+\frac{3}{4}})\equiv L(\mathbb{F}_{n+\frac{3}{4}})italic_L ( blackboard_F start_POSTSUBSCRIPT italic_m ... | We next explore some results which show that if taking the free product with L(ℤ)𝐿ℤL(\mathbb{Z})italic_L ( blackboard_Z ) or ℛℛ\mathcal{R}caligraphic_R preserves elementary equivalence, then under certain conditions, the trichotomy can be improved to a dichotomy. | In direct analog to Theorem 1.1, we have the following conjectured dichotomy for free group factor elementary equivalence: | We record here that the free group factor alternative for elementary equivalence can be obtained in much the same way as in [MunsterLectures, Theorem 5.1] if taking free products with a fixed tracial von Neumann algebra preserves first-order theories. | Although it seems difficult to prove that taking the free product can ever preserve elementary equivalence, in some cases we have some positive results, as in the following proposition. | D |
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