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EL[∫0nvL(r)(d(−B(r))+1{N(r)=0}((Pt−πtL)−(Pt−πtL))dr)|ℱ0]\displaystyle\mbox{E}_{L}\left[\int_{0}^{n}v^{L}(r)\,\Big{(}d(-B(r))+1_{\{N(r)=0\}}\,\big{(}(P_{t}-\pi_{t}^{L})-(P_{t}-\pi_{t}^{L})\big{)}\,dr\Big{)}\,\Big{|}\,{\cal F}_{0}\right] | Given separate premium and valuation technical bases ℬP{\cal B}^{P} and ℬL{\cal B}^{L}, with pure premium rates πtP=Pt\pi_{t}^{P}=P_{t} and πtL\pi_{t}^{L} respectively, (Pt−πtL)(P_{t}-\pi_{t}^{L}) is called the net premium loading or pure premium loading. If (Pt−πtL)>0(P_{t}-\pi_{t}^{L})>0 the loading may be regarded a... | Corollary 6 shows that choosing the valuation basis amounts to choosing a subdivision of the surplus into surplus capitalized at outset as −V0L-V_{0}^{L}, and surplus that emerges later. Proposition 5 (and equation (67)) shows how this also subdivides the premium loadings. The next section completes the relationships s... | To exclude the pure premium loadings Pt−πtLP_{t}-\pi_{t}^{L} from the future premiums valued. That is, in calculating policy values VtLV_{t}^{L} by solving Thiele’s equation backwards from VnL=S¯V_{n}^{L}=\bar{S}, to substitute the pure premium rate πtL\pi_{t}^{L} for the actual premium rate PtP_{t}. | It follows that initial surplus is −V0L-V_{0}^{L}, and all the pure premium loadings are capitalized at outset. Inspection of the proof of Proposition 3 shows that it holds unaltered, since the valuation pure premium rate πtL\pi_{t}^{L} is irrelevant. | D |
We assume that the number of Monte Carlo samples is MM and the number of particle is PnumP_{\mathrm{num}}. Then, with the maximum number of iterations TmaxT_{\max}, the computational complexity of the PSO algorithm for maximizing RskR_{\mathrm{sk}} is approximate to | Figure 2 illuminates the KGR of different benchmarks. From this figure, we can observe that the FA based PLKG with AO algorithm achieves the highest KGR. Therein, AO algorithm consistently improves the KGR and converges to the highest value of approximately 52 bits/s/Hz, which demonstrates the effectiveness of the stag... | Since Monte Carlo method requires multiple sampling and the spatial dimension significantly increases with the substantial increase in the number of antennas, the PSO algorithm shows a high computational complexity in this situation. Therefore, we next consider an AO algorithm to reduce the computational overhead. | This paper studied a novel FA enabled PLKG under the quasi-static wireless environments. We proposed a joint optimization framework based on PSO algorithm that simultaneously adapts the antenna positions and the precoding matrix to maximize the KGR. Moreover, to reduce computational complexity, we developed an AO algor... | We assume that the number of Monte Carlo samples is MM and the number of particle is PnumP_{\mathrm{num}}. Then, with the maximum number of iterations TmaxT_{\max}, the computational complexity of the PSO algorithm for maximizing RskR_{\mathrm{sk}} is approximate to | B |
Then 𝒰α∖𝒰β=∅{\mathcal{U}}_{\alpha}\setminus{\mathcal{U}}_{\beta}=\emptyset, which is finite, so 𝒰{\mathcal{U}} is a | families of covers of XX, then we define the property (𝔘𝔙){{\mathfrak{U}}\choose{\mathfrak{V}}}, read “𝔘\mathfrak{U} choose 𝔙\mathfrak{V}”, as follows: | So for the space (ℵ1,ℵ1∪{ℵ1})(\aleph_{1},\aleph_{1}\cup\{\aleph_{1}\}) we have (ΩT)≠(ΩΓ){\Omega\choose T}\neq{\Omega\choose\Gamma}. | For x∈Xx\in X we set 𝒰x={U∈𝒰:x∈U}{\mathcal{U}}_{x}=\{U\in{\mathcal{U}}:x\in U\} and call 𝒰x{\mathcal{U}}_{x} the | Since we have seen that Γ⊆T\Gamma\subseteq T, property (ΩΓ){\Omega\choose\Gamma} always implies (ΩT){\Omega\choose T}. But the | B |
One of the variants consists of using the RegionPlot3D function in Mathematica to generate graphical representations that help identify whether there are points or areas where the hypotheses are not verified. | For this method, a series of issues arise that make the provided result not always ”certain”. This is determined, for instance, by at least one of the following reasons (the examples are extracted from the computations we have seen on our numerical simulations): | - If for a point (x0,y0,z0)(x_{0},y_{0},z_{0}) we obtain a negative value, then the point is marked as a fail point; In this case the conditions will be evaluated in another 300 points around the point will be evaluated. These points are uniformly distributed, 150 of them being at a distance of 10−810^{-8} and another ... | For the family of energies listed in the next section, we implement numerically the check of the hypothesis from the results presented above and in Section 4. For instance, for the energy considered before in the introduction section just as an example, we check the monotonicity and the invertibility of some stress-str... | However, in order to check the regions on which the local monotonicity and the local invertibility are properly satisfied, we do not use some already implemented function in some math software, but we have implemented our own numerical variants. This is necessary because we have identified some problems which can arise... | A |
We note a few things about this process. First, not all integers are filtered by filters in 𝒮\mathcal{S}. We saved these numbers for later processing and found that none of them were prime, therefore they are accounted for by some earlier prime pp which was filtered out. Second, some filters are more efficient than ot... | We also remark that computer-aided checking of numbers greater than 101710^{17} requires us to work around the integer size limits of most programming languages. We generated R8R_{8} using a Python rewrite of Salez’ algorithm, as the language does not have integer limits, and checked the remaining integers in C++ using... | Salez defined a modular filter SmS_{m} as the set of residue classes mod mm for which the conjecture is known to be true and offered an algorithm to produce these filters. Using modular filters, Salez immediately obtains the Mordell result by applying the Chinese remainder theorem on the identities implied by S5={0,2,3... | We note a few things about this process. First, not all integers are filtered by filters in 𝒮\mathcal{S}. We saved these numbers for later processing and found that none of them were prime, therefore they are accounted for by some earlier prime pp which was filtered out. Second, some filters are more efficient than ot... | We divided work in batches Bk={r+kG8|r∈R8}B_{k}=\{r+kG_{8}\,|\,r\in R_{8}\} for the sake of multithreading. Verifying the conjecture for all primes p≤1018p\leq 10^{18} is equivalent to checking all batches up to k=38641709k=38641709, which can be done in parallel. Additionally, the original 101710^{17} result saves us... | A |
→2φ~0(0)⋅μ0(ℓ)({w∈𝒦:P(ℓ)(w)ij=0})\displaystyle\to 2\widetilde{\varphi}_{0}(0)\cdot\mu^{(\ell)}_{0}(\{w\in\mathcal{K}:P^{(\ell)}(w)_{ij}=0\}) | Since φ~0(0)\widetilde{\varphi}_{0}(0) is the conditional expectation of φ(w)\varphi(w) given P(ℓ)(w)ij=0P^{(\ell)}(w)_{ij}=0: | 2φ~0(0)⋅μ0(ℓ)({w:P(ℓ)(w)ij=0})\displaystyle 2\widetilde{\varphi}_{0}(0)\cdot\mu^{(\ell)}_{0}(\{w:P^{(\ell)}(w)_{ij}=0\}) | Since φ\varphi is compactly supported with ‖φ‖∞≤Cφ\|\varphi\|_{\infty}\leq C_{\varphi} for some constant CφC_{\varphi}: | where φ~εk(z)\widetilde{\varphi}_{\varepsilon_{k}}(z) is the conditional expectation of φ(w)\varphi(w) given P(ℓ)(w)ij=zP^{(\ell)}(w)_{ij}=z and w∈𝒦δ0w\in\mathcal{K}_{\delta}^{0}. | A |
If (Xθ,σ)(X_{\theta},\sigma) is generated by a constant length, left proper substitution θ\theta, then (Xθ,σ)(X_{\theta},\sigma) is a Toeplitz flow. | We now explicitly define the KR-partitions 𝒫(k)\mathcal{P}(k) for Toeplitz flows and substitution systems. | Because it is known that minimal bounded speedups of odometers are conjugate odometers, and Toeplitz flows are symbolic, minimal, almost one-to-one extensions of odometers, it was conjectured in [3] that the minimal bounded speedup of a Toeplitz flow is a Toeplitz flow. We demonstrate that this conjecture is false. We ... | We now generalize Definition 2.17 and Lemma 2.18 from the substitution setting to the more general S-adic setting for Toeplitz flows. | We now define the α\alpha-adic odometer, as there is a close relationship between odometers and Toeplitz flows. | D |
We propose a new construction that improves on this method by processing two stars at a time—a structure we refer to as a double-star. To motivate our approach, consider the graph in Figure 3, called the prototype graph, which is the simplest instance of a double-star with centers at vertices 1 and 2. | The graph coupling number of the prototype graph is gc(G)=6gc(G)=6, and the following pair (𝐏,𝐖)(\mathbf{P},\mathbf{W}) is optimal: | Substituting 𝐏\mathbf{P} and 𝐖\mathbf{W} into the problem formulation verifies that the pair is feasible. To show that six is the minimum number of rows, we perform an exhaustive search over all possible matrices 𝐏∈{±1}5×6\mathbf{P}\in\{\pm 1\}^{5\times 6}. Once 𝐏\mathbf{P} is fixed, the condition 𝐏⊤𝐖𝐏=𝐀+tr(�... | In this work, we focused on the Graph Coupling problem for unweighted graphs. We improved the combinatorial construction of Rajakumar et al. [21], reducing the upper bound on the graph coupling number from 3n−23n-2 to 2.5n+22.5n+2 for any graph with nn vertices. Furthermore, we established the order-optimality of bot... | In Lemma 2, we established a feasible solution (𝐏,𝐖)(\mathbf{P},\mathbf{W}) with six rows for the prototype graph, one of which is the all-ones row. | A |
In the extreme case where ϵ=0\epsilon=0, the gap of W(s)W(s) becomes 0 as expected (numerical gap is ∼10−16\sim 10^{-16}, reaching machine epsilon), while the gap of H(s)H(s) is still ∼10−3\sim 10^{-3}. | Table 3 shows the spectral gaps with different ϵ\epsilon, and we can observe that, as ϵ\epsilon decreases, the gap of H(s)H(s) vanishes while the gap of W(s)W(s) remains at a constant level. | Here we directly choose ϵ=0\epsilon=0, so the Hamiltonian H(s)H(s) is gapless and the walk operator W(s)W(s) of first-order Trotter with time step size 11 still satisfies the gap condition. | Table 2: Spectral gaps of H(s)H(s) and W(s)W(s) in the first example. Here the gap reported is the minimal gap over equi-distant points with step size 10−410^{-4}. | Table 3: Spectral gaps of H(s)H(s) and W(s)W(s) in the second example. Here the gap reported is the minimal gap over equi-distant points with step size 10−410^{-4}. | C |
Furthermore, by Lemma 6.2, the resolvent set of Ah0A_{h_{0}} contains all ω∈ℂ\omega\in\mathbb{C} with Re(ω)>ω0\operatorname{Re}(\omega)>\omega_{0}, where | Using the property of the interpolation space 𝒳α\mathcal{X}_{\alpha} from Definition 3.3 (2), we obtain the following estimate for the first term in (7.1). | we define 𝒳α:=(𝒳0,𝒳1)α=𝔥λ,s2α+ρ\mathcal{X}_{\alpha}:=(\mathcal{X}_{0},\mathcal{X}_{1})_{\alpha}=\mathfrak{h}_{\lambda,s}^{2\alpha+\rho}, which represents the continuous interpolation space between 𝒳0\mathcal{X}_{0} and 𝒳1\mathcal{X}_{1}. The precise definition is provided in Definition 5.1. | We will prove the following stability result for cusped hyperbolic 3-manifolds using the interpolation theory. | There exists θ∈(0,1)\theta\in(0,1), such that the following statement is true. Denote by (ℰ0,D(A~))θ(\mathcal{E}_{0},D(\tilde{A}))_{\theta} the continuous interpolation space. And define the following set | A |
Table 2: The coefficient values relating the explanatory variables to the main response variable (diagnosis of mesothelioma) obtained with naive mle and G-hive in the lung cancer dataset. | \sayplatelet count to be the two most important variables in classifying whether a patient has mesothelioma or not, and this is consistent with the findings with our method, even in the setting of having the asbestos related variables considered hidden confounders and removed. In terms of magnitude, the coefficient val... | Next we evaluate the estimation error of Θ^\hat{\Theta} in three scenarios: (1) when we vary the level of influence of the hidden variables through the magnitude of η\eta, (2) when we increase the sample size nn, and (3) when we increase the dimension of the response MM. For (1) we vary η\eta to take values in {1,2,…,8... | As it is impossible to know the ground truth, we rely on the results provided in Chicco and Rovelli (2019) to gauge the accuracy of our method. The authors in Chicco and Rovelli (2019) conclude \saylung side and | We apply our data driven G-hive procedure to a dataset from Chicco and Rovelli (2019) regarding mesothelioma, a type of lung cancer. Specifically, this dataset consists of real electronic health records on 324 patients in Turkey of which 96 are diagnosed with mesothelioma and 228 are not. The dataset includes 33 explan... | C |
Furthermore, given k∈ℕk\in\mathbb{N}. For any D>0D>0, there exists d=d(M,h0,D,k)≤min{d0,D}d=d(M,h_{0},D,k)\leq\min\{d_{0},D\} with the following property. | In this section, we review the stability result associated with the normalized Ricci-DeTurck flow. It is shown in [5] that, under C0C^{0} perturbations of the hyperbolic metric h0h_{0}, the corresponding flow exists for all time and remains close to h0h_{0}. The following result is deduced from [5, Theorem 1.1] in [29,... | The paper is organized as follows. Section 2 discusses the equidistribution properties of minimal surfaces with respect to the hyperbolic metric and establishes Theorem A. In Section 3, we examine conditions on general metrics on MM that ensure the existence of minimal surfaces. Theorem B is proved in Section 4. Sectio... | In [29], the authors provided a more quantitative version of the stability of cusped hyperbolic manifolds under normalized Ricci-DeTurck flow. We impose additional conditions on the initial metric and use Bamler’s stability result [5] to rule out trivial Einstein variations. The strategy builds on maximal regularity th... | In this section, we review the long time behavior of the normalized Ricci-DeTurck flow and its convergence toward the hyperbolic metric. In particular, we present a quantitative exponential decay estimate, which plays an essential role in the proofs of Theorems C and D. | D |
Can contemporary mathematics be reconciled with the classical quadrivium? Whether we examine linear K-12 school curricula111Reflected in, for example, the Common Core Mathematics standards at corestandards.org. in the U.S., which try to impose a hierarchy upon loose clusters of topics, or the research triumvirate of an... | The long and varied history of algebra provides ample resources to substantiate this view of its fundamental character. Four key periods illustrate the role algebra plays in the liberal arts: the era preceding Viète’s Introduction, the early years of algebraic geometry (also called analytic geometry), a transition peri... | Recall that in Boethius’ language,666His original description in the Proemium of the De Institutione Arithmetica [c. 500 A.D.] is well worth meditating on. A translation can be found in Michael Masi, Boethian Number Theory (New York: Rodopi, 1983), 72. the quadrivium consists of arithmetic (concerning number in its abs... | Just as Leonhard Euler became the inheritor of Gottfried Leibniz’s calculus, so too did he receive and transform the algebraic tradition. Many of his early works elaborate what later became known as number theory,454545C. Edward Sandifier, The Early Mathematics of Leonhard Euler (Providence: MAA Press, 2007) is full of... | Can contemporary mathematics be reconciled with the classical quadrivium? Whether we examine linear K-12 school curricula111Reflected in, for example, the Common Core Mathematics standards at corestandards.org. in the U.S., which try to impose a hierarchy upon loose clusters of topics, or the research triumvirate of an... | B |
Poisson:Δu=fonΩ, and u=gon∂Ω.\displaystyle\text{Poisson:}\quad\Delta u=f\quad\text{on}\quad\Omega\text{, and }\quad u=g\quad\text{on}\quad\partial\Omega. | In general, our WoS-NN method follows three main steps: the stochastic representation of PDEs, discretization of stochastic processes, and neural network approximations. The novel contributions of our method are as follows: | In section 2, a background on elliptic PDE solvers is provided, including the original WoS method and recent machine-learning approaches in PDE solving. In Section 3, we present the primary processes of the WoS-NN method, including the mathematical background and our neural network design. Section 4 compares WoS-NN wit... | Solving these two equations with our WoS-NN method is crucial, as they are the foundation of solving other elliptic PDEs. Extensive experiments and detailed analyses of these two equations have demonstrated the significant advantages of our method over the original Walk on Spheres method and other stochastic approaches... | In the modern development of science and engineering, partial differential equations (PDEs) are the basis of various natural phenomena and industrial applications. Solving elliptic PDEs such as Laplace and Poisson equations enhances our understanding of natural processes and drives advancements across multiple industri... | C |
The ultrametric spaces (𝐁¯X0,dH)(\bar{\mathbf{B}}_{X}^{0},d_{H}) and (𝐁¯Y0,ρH)(\bar{\mathbf{B}}_{Y}^{0},\rho_{H}) are isometric. | Theorem 5.7 gives sufficient conditions under which (X,d)(X,d) and (𝐁¯X,dH)(\bar{\mathbf{B}}_{X},d_{H}) are isometric. The necessary and sufficient conditions under which the space (𝐁¯X,dH)(\bar{\mathbf{B}}_{X},d_{H}) is separable proved in Theorem 5.17. Example 5.18 describes a separable ultrametric space (X,d)(X,d)... | Theorem 5.7 gives us the condition under which (X,d)(X,d) and (𝐁¯X,dH)(\bar{\mathbf{B}}_{X},d_{H}) are isometric for equidistant (X,d)(X,d). The next question naturally arises. | The next our goal is to describe conditions under which (𝐁¯X,dH)(\bar{\mathbf{B}}_{X},d_{H}) is separable. | The following lemma will be used to describe the condition under which the ballean 𝐁¯X\bar{\mathbf{B}}_{X} is complete. | B |
ℓReλ=−t+e−t(pcoss+qsins) and ℓReλ=t+et(pcoss−qsins).\ell\,\operatorname{Re}\lambda=-t+e^{-t}(p\cos{s}+q\sin{s})\;\text{ and }\;\ell\,\operatorname{Re}\lambda=t+e^{t}(p\cos{s}-q\sin{s}). | Using the first identity for t≥0t\geq 0 and the second identity for t≤0t\leq 0, we immediately infer (3.16). Similarly, from (3.19) and (3.21) we obtain the identities | In the sequel we will use the following identity satisfied by the matrix-values function W(⋅)W(\cdot) and the Jost solutions U±(⋅,⋅)U_{\pm}(\cdot,\cdot), | We recall the following identity for the algebraic multiplicity of matrices obtained by functional calculus: | and assertion (3.17) follows by using the first identity for t≥0t\geq 0 and the second for t≤0t\leq 0. | A |
In his 1963 paper, Moser [7] proved that the average value of f(n)f(n) is asymptotically equal to log2\log 2; that is, | For these shorter chains, 𝐬𝐥𝐢𝐝𝐞{\mathbf{slide}} operations take, on average, O(logx2)O(\log x_{2}) | Several different practical implementations of the algorithms presented in §2 and §3 were created and run over several upper bounds, and we present both their results and timing information below. Values of the histogram function h(k,x)h(k,x), where xx ranges over powers of ten111The authors are aware that Tables 1 an... | These questions are then reprised in problem C2 from [5], and several Online Encyclopedia of Integer Sequences (OEIS) entries address these questions: | In particular interest to Moser’s second and fourth questions, OEIS A054859 lists the smallest integer having exactly kk representations. Prior to our work, this sequence was known to k=13k=13; we present the first known integer with 14 representations as a sum of consecutive primes. Table 4 gives the complete updated ... | C |
2) There is no such λ∈𝔽\lambda\in{\mathbb{F}} that wi=λviw_{i}=\lambda v_{i} for all i=1,…,mi=1,\ldots,m. | 2) There is no such λ∈𝔽\lambda\in{\mathbb{F}} that wi=λviw_{i}=\lambda v_{i} for all i=1,…,mi=1,\ldots,m. | Now we consider generators of an incidence algebra 𝒜=𝒜n(⪯,ℝ){\mathcal{A}}={\mathcal{A}}_{n}(\preceq,{\mathbb{R}}) over the field ℝ{\mathbb{R}} of real numbers. In this case, the direct sum 𝒜m{\mathcal{A}}^{m} can be viewed as the arithmetic vector space ℝmρ{\mathbb{R}}^{m\rho}, ρ=dimℝ𝒜\rho=\dim_{\mathbb{R}}{\math... | 1) ⇒\Rightarrow 2) The direct substitution wi=λviw_{i}=\lambda v_{i} into the system results in contradiction. | 2) ⇒\Rightarrow 1) Consider the outer direct sum Vm=V⊕…⊕VV^{m}=V\oplus\ldots\oplus V (mtimes)(m~\text{times}). For i=1,…,mi=1,\ldots,m, we denote the i-th direct summand by ViV_{i}, the projection by πi:Vm⟶Vi{\pi_{i}:V^{m}{\longrightarrow}V_{i}}, and the natural embedding by ιi:Vi⟶Vm\iota_{i}:V_{i}{\longrightarrow}V^{... | C |
𝒥(ℋℓn)∼𝒥(ℋℓn−1)×B{\mathcal{J}}({\mathcal{H}}_{\ell^{n}})\sim{\mathcal{J}}({\mathcal{H}}_{\ell^{n-1}})\times B | Write d=ℓnd=\ell^{n}. In characteristic 0, this BB has complex multiplication with endomorphism algebra | generating (ℤ/ℓnℤ)×(\mathbb{Z}/\ell^{n}\mathbb{Z})^{\times}; write m=ℓn−1(ℓ−1)m=\ell^{n-1}(\ell-1) for the order of this group. | contains the CM field Kℓ:=ℚ(i,ζℓ+ζℓ−1)K_{\ell}:=\mathbb{Q}(i,\zeta_{\ell}+\zeta_{\ell}^{-1}) in its endomorphism algebra; | Write d=ℓ1ℓ2d=\ell_{1}\ell_{2}. In characteristic 0, this AA has complex multiplication with endomorphism algebra | A |
Again the control input at the root of the beam causes the ripples while stabilizing the torsion of the beam. | In the above sections we have constructed a full state feedback to asymptotically stabilize the linear system. But we don’t have the ability to measure the full state so need to construct a Kalman filter to deliver an estimate of the full state from a finite number of point measurements. We assume that we can measure | that processed two point measurements at the tip of the beam to obtain an estimate of the full state of the beam. This yields an Linear | We extend LQG to designing a dynamic compensator for a controlled and observed infinite dimensional dynamical system under point actuation and point sensing. The LQG approach breaks the problem into two parts. The first part uses Linear Quadratic Regulator theory to find a stabilizing linear feedback. This feedback ass... | We expect that the state z(y1,t−s)z(y_{1},t-s) in the far past s>>0s>>0 to be irrelevant to the estimate z^(y1,t)\hat{z}(y_{1},t) of the state at time tt so we assume that ℋ(y,y1,s)z(y1,t−s)→0{\cal H}(y,y_{1},s)z(y_{1},t-s)\to 0 as s→∞s\to\infty. Hence | A |
The class of planar graphs is closed under taking minors, and the class of string graphs is closed under taking induced minors [20]. Explicitly: | If GG is a planar graph, then any graph obtained from GG by a sequence of vertex deletions, edge deletions, and edge contractions is also a planar graph. | If GG is a string graph, then any graph obtained from GG by a sequence of vertex deletions and edge contractions (but not edge deletions) is also a string graph. | Let GG be a graph with maximum degree three. Then GG is a string graph if and only if there exists a matching MM of GG such that the graph obtained from GG by contracting MM is planar. | If a graph GG has a matching MM such that the contraction of the matched edges, G/MG/M, is planar, then GG is a string graph. | A |
1|r−r′|=4π∑l=0∞∑m=−ll12l+1r<lr>l+1Ylm∗(θ′,ϕ′)Ylm(θ,ϕ),\displaystyle\frac{1}{|{\bi r}-{\bi r}^{\prime}|}=4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\frac{1}{2l+1}\frac{r_{<}^{l}}{r_{>}^{l+1}}\,Y_{lm}^{*}(\theta^{\prime},\phi^{\prime})Y_{lm}(\theta,\phi), | For n>0n>0, the integral in (A6) can be evaluated in terms of the Appell function F1F_{1} and the hypergeometric function F12{}_{2}F_{1}. As ϵ→0\epsilon\rightarrow 0, the arguments of both functions approach ∞\infty; their | where Jm(⋅)J_{m}(\cdot) are the Bessel functions of the first kind of order mm and z>z_{>} (z<z_{<}) is the greater (lesser) of the zz-coordinates of the vectors r\bi r and r′{\bi r}^{\prime}. | 1/r1/r at r→∞r\rightarrow\infty, and thus the integral representation of the solution does not converge. | using the standard integral representation of its solution, the operator of which we term the inverse Laplacian and denote by Δ¯−1\bar{\Delta}^{-1}. Thus | C |
We now multiply the inequality above by m(x0,𝐁)ℓ+d−1m(x_{0},\mathbf{B})^{\ell+d-1} and integrate the resulting inequality in x0x_{0} | Using |𝐁(x)|≤Cm(x,𝐁)2|\mathbf{B}(x)|\leq Cm(x,\mathbf{B})^{2} and |∇𝐁(x)|≤Cm(x,𝐁)3|\nabla\mathbf{B}(x)|\leq Cm(x,\mathbf{B})^{3}, we obtain | |𝐁(x)|1/2+|∇𝐁(x)|1/3≤Cm(x,𝐁)|\mathbf{B}(x)|^{1/2}+|\nabla\mathbf{B}(x)|^{1/3}\leq Cm(x,\mathbf{B}) | Since |𝐁(x)|≤Cm(x,𝐁)2|\mathbf{B}(x)|\leq Cm(x,\mathbf{B})^{2} and |∇𝐁(x)|≤Cm(x,𝐁)3|\nabla\mathbf{B}(x)|\leq Cm(x,\mathbf{B})^{3}, we obtain | Using the fact that m(x,𝐁)≈m(y,𝐁)m(x,\mathbf{B})\approx m(y,\mathbf{B}) if |x−y|<cm(x,𝐁)−1|x-y|<cm(x,\mathbf{B})^{-1}, | D |
In this section we discuss the superconvergence of the discontinuous Galerkin approximation to Equation eq. 1. We begin by introducint the superconvergent points in the one-dimensional steady-state case. We then proceed to discuss the multi-dimensional time-dependent case and the underlying superconvergence in the nega... | Potential future directions are as follows: (1) For varying cross sections, our current superconvergence proof can be extended by establishing divided difference results for the DG approximation. Though numerically observed superconvergnce, a complete superconvergence proof for spatial cross sections is still an open q... | On the other hand, following [20], to help the error analysis in the time, since the cross-sections σa\sigma_{a} and σt\sigma_{t} are constant in space, we can reduce the analysis to Equation (1a) to a purely scattering system for the function (abusing notation) | equipped with proper boundary conditions. Here, ψ(𝒙,𝛀,t)\psi(\boldsymbol{x},\boldsymbol{\Omega},t) is the particle distribution function (also known as intensity or angular flux) for spatial location 𝒙∈𝑿⊂ℝ3\boldsymbol{x}\in\boldsymbol{X}\subset\mathbb{R}^{3}, angular direction on the unit sphere 𝛀∈𝕊2\boldsymbol{... | In our superconvergence analysis, we assume periodic or zero inflow boundary conditions as well as a constant scattering cross section, σs\sigma_{s}. Note that though our analysis is restricted to constant scattering cross sections, we numerically observe improved accuracy after post-processing for general cross sectio... | D |
𝒞(𝒫,D):={(ϕ(P1),…,ϕ(Pn)):ϕ∈ℒE(D)}.\displaystyle\mathcal{C}(\mathcal{P},D):=\{(\phi(P_{1}),\dots,\phi(P_{n})):\;\phi\in\mathcal{L}_{E}(D)\}. | Then 𝒞(𝒫,D)\mathcal{C}(\mathcal{P},D) is a linear code with dimension ℓ(D)\ell(D) and minimum distance at least n−deg(D)n-\deg(D). | is a finite-dimensional vector space over 𝔽q\mathbb{F}_{q}. We denote its dimension by ℓ(D):=dim𝔽qℒ(D)\ell(D):=\dim_{\mathbb{F}_{q}}\mathcal{L}(D), which is at least deg(D)+1−g(E)\deg(D)+1-g(E) by Riemann’s theorem (see [33, Theorem 1.4.17]). If deg(D)≥2g(E)−1\deg(D)\geq 2g(E)-1, then it holds | Consequently, 𝒞(𝒫,V)\mathcal{C}(\mathcal{P},V) is an [n,k,d]q[n,k,d]_{q}-linear code with dimension k=dim𝔽q(V)k=\dim_{\mathbb{F}_{q}}(V), and its minimum distance dd remains at least n−deg(D)n-\deg(D). | Hence, 𝒞(𝒫,V)\mathcal{C}(\mathcal{P},V) has dimension tr+1tr+1 and minimum distance at least (m−t)N(m-t)N by Section II-A. By the Singleton-type bound (1), we conclude that the minimum distance of 𝒞(𝒫,V)\mathcal{C}(\mathcal{P},V) is exactly (m−t)N(m-t)N, and 𝒞(𝒫,V)\mathcal{C}(\mathcal{P},V) is an optimal (r... | A |
Theorem 2 (Theorem 3.6). Let GG be a hierarchically hyperbolic group. Then, the hierarchical boundary of GG is connected if and only if GG is one-ended. | It is well known that the Gromov boundary of a hyperbolic group is connected if and only if it is one-ended. The same is true for the Bowditch boundary of a relatively hyperbolic group [7, Theorem 10.1]. So it is natural to look for the relationship between the hierarchical boundary of a hierarchically hyperbolic group... | In [13], Hamenstädt introduced a ZZ-boundary of the mapping class group of a surface of finite type. In Lemma 3.2, we show that this ZZ-boundary is weakly visible, and hence by Theorem 1, it is connected (Proposition 3.3). As an application of Theorem 1 and [4, Theorem 1.3], we obtain the following, and answer a questi... | For the definition of a hierarchically hyperbolic group and its boundary, one is referred to Section 2. In [10, p. 3672], the authors conjectured Theorem 2. Here, we prove their conjecture. An application of Theorem 2 implies that the hierarchical boundary of the mapping class group of a connected orientable surface of... | Hierarchical boundary. In [10], the authors introduced the notion of a boundary of an HHS. From [10, Section 2], we recall the definition of the hierarchical boundary and its topology. For S∈𝔖S\in\mathfrak{S}, ∂𝒞S\partial\mathcal{C}S denote the Gromov boundary [11] of 𝒞S\mathcal{C}S. | C |
We first prove the following inequality involving non-tangential maximal function Mℒ∗M^{*}_{\mathcal{L}} and grand maximal function ℳ\mathcal{M} (defined in Section 3). | Due to the connection with the Heisenberg group, there is an induced twisted convolution on ℝ2n{\mathbb{R}}^{2n} with which the operator ℒ\mathcal{L} commutes. | Before proving Theorem 3.1, let us first state a useful lemma we will be using throughout this paper. | Before proving (3.12), let us first derive a useful relation between twisted convolution and convolution on Heisenberg group when restricted to particular type of functions. | Recall that τwf(z)=f(z−w)ei2Im(z.w¯)\tau_{w}f(z)=f(z-w)e^{\frac{i}{2}Im(z.\bar{w})} is the twisted translation of ff by ww. In fact, τw\tau_{w} defined above is indeed the left translation by (−w,0)(-w,0) on the Heisenberg group restricted on the functions of type f(z)eitf(z)e^{it} and evaluated at t=0t=0. | C |
W(T,S)={σ∈ℂ:‖xn‖=1,‖Txn‖→‖T‖and⟨Txn,Sxn⟩→σ}.W(T,S)=\big{\{}\sigma\in\mathbb{C}:\|x_{n}\|=1,\|Tx_{n}\|\to\|T\|\,\,\mbox{and}\,\langle Tx_{n},Sx_{n}\rangle\to\sigma\big{\}}. | [5] Let T∈𝔹A1/2(ℍ)T\in\mathbb{B}_{A^{1/2}}(\mathbb{H}) and R(A)R(A) be closed. Then TT is AA-compact if and only if T~\widetilde{T} is compact. | In the following theorem, we completely characterize the AA-smoothness of AA-bounded operators provided that MA(T)M_{A}(T) is nonempty. This will lead to the characterization of the AA-smoothness of AA-compact operators. | Now we establish that AA-compact operators possess a nonempty AA-norm attainment set whenever R(A)R(A) is closed. | In this connection we observe a relation between the AA-norm attainment set of T∈𝔹A1/2(ℍ)T\in\mathbb{B}_{A^{1/2}}(\mathbb{H}) and the norm attainment set of T~∈𝔹(R(A1/2)).\widetilde{T}\in\mathbb{B}(\textbf{R}(A^{1/2})). | C |
Some patterns of penalty functions for SigL (panel A), ASigL (panel B), SL_prod (panel C), SL_min (panel D) in two dimensional case (p=2) are given in Fig. 1, respectively. The first column represents the contour for penalty function of four methods, the second column represents the penalty function versus β\beta, and ... | In this paper we propose a new version of signal lasso based on two kinds of penalty function to estimate the signal parameter and uncovering network topology in complex network with a small amount of observations. We find the tuning parameter can be set to a large enough values such that the signal parameter can be co... | Although lasso method, have ability of shrink the parameter estimates toward to zero under the natural sparsity in complex network, the existent links between nodes can not be shrunk to its true value of 1, which will inevitably decrease the estimation accuracy in most cases. For this reason, Shi et al. (2021) proposed... | Comparing with the additive penalty in signal lasso or adaptive signal lasso, these two penalties are not convex function, however they have appealing property that just one tuning parameter be involved and the βj\beta_{j} can be shrunk to either 0 or 1 completely as shown later. We will prove that there is no need to ... | It is clear that SigL method (panel A) fails to shrink the values of β^\hat{\beta} that fall in the interval (0.1,0.9) to the correct class and leave an unclassified portion in network reconstruction. ASigL method improve this weakness such that the middle part between 0 and 1 can be shrunk toward to two directions. Wh... | D |
Without loss of generality, we use ηsat\eta_{\mathrm{sat}} to denote the minimum coverage ratio of the LEO satellite IoT constellation in all time slots, i.e., | Figure 4: Division of Earth’s surface using the grid method in STK with a grid precision of 10∘10^{\circ}. | In above, we obtained the coverage area of a LEO satellite. Actually, we are more concerned about the coverage ratio of the LEO satellite IoT constellation. To calculate the coverage ratio ηsat\eta_{\mathrm{sat}} of the target area served by the LEO satellite IoT constellation, the grid method is adopted as illustrated... | Based on the above analysis, the calculation of coverage ratio is related to the grid division precision and time division. Specifically, the coverage ratio is more accurate with higher grid precision and shorter time steps. However, increasing grid precision and shortening time steps inevitably increase computation ti... | Based on the location of a satellite under Cartesian coordinates in (1) and the coverage area of a satellite in (5), we can indicate whether an observation grid is covered as conducted in (6). Then, the coverage ratio of a LEO satellite IoT constellation in time slot tt can be calculated as | C |
Among them, Jacobsson and Löfwall’s one [JL91, Definition 3.4] and Coquand and Persson’s one [CP99, Section 3.1] are (generalized) inductive definitions. | In Section 2, based on the inductive definitions of Noetherianity [JL91, CP99], we define the notion of α\alpha-Noetherian modules and α\alpha-Noetherian rings for an ordinal α\alpha. | In Section 3, we prove a quantitative version of Hilbert’s basis theorem (Theorem 3.7): if a ring AA is α\alpha-Noetherian, then A[X]A[X] is (ω⊗α)(\omega\otimes\alpha)-Noetherian, where ⊗\otimes denotes the Hessenberg natural product. | The above definition of α\alpha-Noetherian modules is a quantitative version of the following generalized inductive definition of Noetherian modules: | In this paper, we quantify the inductive definition and define α\alpha-Noetherian rings for an ordinal α\alpha. Then we constructively prove a quantitative version of Hilbert’s basis theorem and present an application to the Krull dimension of polynomial rings. | D |
A geodetic set of a graph GG is a set SS of vertices such that any vertex of GG lies on some shortest path between two vertices of SS [GS]. The geodetic number g(G)g\left(G\right) of GG is the smallest possible size of a geodetic set of GG. | The version where the edges must be covered is called an edge-geodetic set [EGS]. “Strong” versions of these notions have been studied. A strong (edge-) geodetic set of graph GG is a set SS of vertices of GG such that we can assign for any pair x,yx,y of vertices of SS a shortest xyxy-path such that each vertex (edge)... | We will first show the bound meg(G)≤3c(G)+ℓ(G)+1\mathrm{meg}\left(G\right)\leq 3c\left(G\right)+\ell\left(G\right)+1. We say that an edge ee is monitored by a set SS if there are two vertices x,yx,y of SS such that ee lies on all isometric paths between xx and yy; SS is thus a monitoring edge-geodetic set of GG if... | Recently, the concept of monitoring edge-geodetic set was introduced in [MEG1] (see also [MEGbilo2024inapproximability, dev2023monitoring, MEG-CALDAM2024, MEG2]) as a strengthening of a strong edge-geodetic set: here, for every edge ee, there must exist two vertices x,yx,y in the monitoring edge-geodetic set such that... | A geodetic set of a graph GG is a set SS of vertices such that any vertex of GG lies on some shortest path between two vertices of SS [GS]. The geodetic number g(G)g\left(G\right) of GG is the smallest possible size of a geodetic set of GG. | A |
For this case, it is necessary to impose an additional condition to guarantee the square-integrability of the states described in Equation (99), which is given by | The presence of a critical electric field strength, defined by ℰ0=ℬ0(vy−νvt)\mathcal{E}_{0}=\mathcal{B}_{0}(v_{y}-\nu v_{\rm t}), plays a crucial role. As the electric field approaches this critical value, all energy levels collapse to the unique value E=−kvdE=-kv_{\mathrm{d}}, indicating a degeneracy point consiste... | Thus, in addition to having a finite number of states due to the condition in Equation (81), we obtain a well-defined domain in kk for each energy level. Consequently, the resulting spectrum is discrete, finite, dispersive, and bounded. However, although the bounds remain linear in kk, they do not exhibit the same slop... | This condition causes the energy spectrum to become discrete but finite, with the maximum number of excited states determined by the fulfilment of the previous condition. The spectrum exhibits a linear behavior in kk with a slope of −vd-v_{d} for the ground state, whereas it is nonlinear for the excited states. Moreove... | Across the different scenarios studied—from constant and exponential profiles to hyperbolic field configurations—we observed a consistent set of behaviors and phenomena. In all cases, the energy spectrum features a term with linear dependence on the wave number kk in the yy-direction. For the ground state, the slope of... | B |
Proof. We already know eα≤ψα(xi)≤xie_{\alpha}\leq\psi_{\alpha}(x_{i})\leq x_{i} for i=1,2,…i=1,2,\ldots | First, the random variable Yi=(X−xi)+Y_{i}=(X-x_{i})_{+} converges pointwise to (X−x¯)+(X-\bar{x})_{+} and is bounded since | Consequently, the sequence {xi}i=0,1,…\{x_{i}\}_{i=0,1,\ldots} converges to some x¯\bar{x} with eα(X)≤x¯≤max{x0,x1}e_{\alpha}(X)\leq\bar{x}\leq\max\{x_{0},x_{1}\}. It remains to verify x¯=ψα(x¯)=limi→∞ψα(xi)\bar{x}=\psi_{\alpha}(\bar{x})=\lim_{i\to\infty}\psi_{\alpha}(x_{i}). | limi→∞𝔼Yi=limi→∞𝔼(X−xi)+=𝔼(X−x¯)+.\lim_{i\to\infty}\mathbb{E}Y_{i}=\lim_{i\to\infty}\mathbb{E}(X-x_{i})_{+}=\mathbb{E}(X-\bar{x})_{+}. | Similarly, the bounded random variables 1IX>xi\mathrm{1\negthickspace I}_{X>x_{i}}, 1IX≤xi\mathrm{1\negthickspace I}_{X\leq x_{i}} converge pointwise to 1IX>x¯\mathrm{1\negthickspace I}_{X>\bar{x}}, 1IX≤x¯\mathrm{1\negthickspace I}_{X\leq\bar{x}}, respectively. Again by the DCT | A |
In the rest of this paper we formulate the stability constrained optimal loading problem, examine its properties and present some numerical calculations for a representative vessel. | We present the results of our calculations in Table 3. In all cases the quadratic constraint is binding. If it were not, the solution would be a linear programming one with only two nonzero cargo types, as many as the Volume and Deadweight constraints. This does not hold for Normal Loading with binding metacentric cons... | The cargo types appear in Table 2 and are chosen so that they give interesting numerical results. For cargo types i=1,2,3,4i=1,2,3,4 the table gives the freight rates in monetary units per metric ton and density in tons per cubic meters. Water density ρ\rho is assumed to be one, thus cargo densities are relative to wat... | We consider a vessel whose operator can select at will quantities from several types of cargoes in order to maximize revenue for a single voyage. Each type is characterized by its density and its freight rate. Loading constraints that have been taken into account in standard treatments [2] are volume and weight ones, a... | We assume that the available cargo types are indexed by ii, have densities did_{i}, and freight rates pip_{i} monetary units per ton. The vessel has a loading capacity of CC tons (Deadweight), while the volume of loaded cargo can not exceed a maximum quantity VV. Assuming that the operator can select any nonnegative ra... | D |
In section 4, we compute the restricted cohomology and restricted one-dimensional central extensions of | extensions of 𝔤\mathfrak{g} by a one-dimensional space 𝔽c\mathbb{F}c. By Theorem 2.3, these restricted one-dimensional central extensions can be determined by restricted 2-cohomology. | We only consider central extensions obtained by even cocycles, because the central extensions obtained by the odd ones are trivial. | ordinary Lie superalgebra extensions and the Lie algebras (𝔜k,l)0¯(\mathfrak{Y}_{k,l})_{\bar{0}} cannot be obtained from the central extensions of Heisenberg Lie algebras [11]. | in central extensions, we only consider trivial coefficients and the full description can be found in | B |
Let KK be a field of characteristic three and let ℰ{\mathcal{E}} be an elliptic curve defined over KK. Its Weierstrass normal form (WNF, for short) can be (see, e.g. [Sil09]) either | which implies γ(x)=c0.\gamma(x)=c_{0}. We get that η=−1x+c0,\eta=-\frac{1}{x}+c_{0}, that is, the formal isogenies is in this case are of the form | Since the neutral element is the point at infinity, Ω\Omega, of homogeneous coordinates [x,y,z]=[0,1,0][x,y,z]=[0,1,0], we see that the inverse of a point P(x,y)P(x,y) is (x,−y)(x,-y) and, as isogenies are group morphisms, we see that f1f_{1} is fact a rational function on x,x, say η(x).\eta(x). As in both cases, the... | Notice that a curve in WNF2 is automatically non-supersingular (cf e.g. [Sil09], Thm. 4.1, pp 148), so the real case of interest is the WNF1; throughout this paper, we will focus on this case only. | Remark 1. It is straightforward that the solution γ\gamma of 8 is usually just a formal power series, even if the function ψ\psi is rational. An easy example is given by the case when γ3+γ=X3\gamma^{3}+\gamma=X^{3}, for which | C |
From now on assume that G=GmG=G_{m}. We recall the explicit formulation of Jacquet-Shalika integrals following [JS90, CM15]. | If ξ\xi is of Whittaker type, then ξ~\tilde{\xi} is of Whittaker type as well and π(ξ~)≅π(ξ)∨\pi(\tilde{\xi})\cong\pi(\xi)^{\vee} by the properties of MVW involution (([MVW87])). | Recall the right action of S2n−1S_{2n-1} on 𝕜n−1\mathbbm{k}^{n-1} from (2.12) and the involution in (3.2). It can be verified that | Since the element τm\tau_{m} given by (2.10) is fixed by the MVW involution h↦h−1th\mapsto{}^{t}h^{-1} on GmG_{m}, the involution Ad(τm){\rm Ad}(\tau_{m}) and the MVW involution commutes. We introduce the following involution | By MVW involution, I(ξ~)I(\tilde{\xi}) has an irreducible generic quotient π(ξ~)≅π(ξ)∨\pi(\tilde{\xi})\cong\pi(\xi)^{\vee}, such that π(ξ~)⊗|η|12\pi(\tilde{\xi})\otimes|\eta|^{\frac{1}{2}} is nearly tempered. By Theorem 2.2 (4) and that L(1−s,π(ξ~),∧2⊗η)\operatorname{L}(1-s,\pi(\tilde{\xi}),\wedge^{2}\otimes\eta) i... | C |
∑n≥0γn(m)(q,1)xn=11−qxℬm(x)\sum_{n\geq 0}\gamma^{(m)}_{n}(q,1)x^{n}=\frac{1}{1-qx\mathcal{B}_{m}(x)} | γn(m)(q,t)\gamma^{(m)}_{n}(q,t) is a linear combination of complete homogeneous polynomials, hi(q,t)h_{i}(q,t) for 0≤i≤n0\leq i\leq n. | ∑1≤i≤k|{j:aj=i}|≥kfor all 1≤k≤n\sum_{1\leq i\leq k}|\{j:a_{j}=i\}|\geq k\ \text{for all}\ 1\leq k\leq n | If 𝐩\mathbf{p} is a parking distribution on a mm-regular caterpillar tree TT, then the number of lucky drivers and the number of drivers that prefer to park at node 11 form a symmetric joint distribution. In fact, the polynomial γn(m)(q,t,1,1)/qt\gamma^{(m)}_{n}(q,t,1,1)/qt is a linear combination of complete homoge... | Furthermore, we have derived a q,tq,t-analog for the Fuss-Catalan numbers and established a linear combination of complete homogeneous polynomials in q0,…,qmq_{0},\dots,q_{m} as a closed form expression for γn(m)(q0,…,qm)\gamma_{n}^{(m)}(q_{0},\dots,q_{m}). | A |
We gratefully acknowledge support from PRIN 2022A7L229 “Algebraic and topological combinatorics”, PRIN 2022S8SSW2 “Algebraic and geometric aspects of Lie theory”, the MIUR Excellence Department Project awarded to the Department of Mathematics of the University of Pisa (CUP I57G22000700001), and INdAM-GNSAGA. | Any reflection group admits a Coxeter presentation by taking as generating set SS the reflections across the hyperplanes bounding any fixed chamber. | Let WW be a Coxeter group and SS its generating set of reflections (usually called simple reflections). | The interval consisting of all vertices lying on shortest paths from 11 to ww in the (right) Cayley graph of WW with respect to the generating set RR is called the (generalized) noncrossing partition poset associated with the Coxeter group WW and the Coxeter element ww (as usual, it is understood that WW is equipped wi... | Denote by RR the set of all reflections of WW; algebraically, RR can be defined as the set of all conjugates of the simple reflections. | B |
Since δ>1\delta>1 and therefore by (*) β<1\beta<1 we see that δ−β\delta-\beta is positive and hence D=F−G≥0D=F-G\geq 0. Moreover, it follows that τ\tau is a maximizing point of DD with D(τ)>0D(\tau)>0. This shows that FF and GG lie in the alternative H1H_{1}. If in addition GG is strictly increasing in a neighborhood ... | s:=(r+k)nn+m∈ℕ∪{0} and s≤min{r,n}s:=(r+k)\frac{n}{n+m}\in\mathbb{N}\cup\{0\}\;\text{ and }\;s\leq\min\{r,n\} | To motivate our test-statistic we investigate TnmT_{nm} under the usual limiting regime, that is min{n,m}→∞\min\{n,m\}\rightarrow\infty and | Rnm:=min{1≤i≤n+m:Fn(Z(i))−Gm(Z(i))=Dnm}R_{nm}:=\min\{1\leq i\leq n+m:F_{n}(Z_{(i)})-G_{m}(Z_{(i)})=D_{nm}\} | In the following theorem we use the notation a∧b:=min{a,b}a\wedge b:=\min\{a,b\} for reals aa and bb. | D |
∂1G1−∂2G2=0 in 𝔻,∂2G1+∂1G2=0 in 𝔻,G1=ReG=g on ∂𝔻\displaystyle\begin{aligned} \partial_{1}G_{1}-\partial_{2}G_{2}&=0&\quad&\textup{~in~}{\mathbb{D}},\\ | ww, while the mean value property for ReG\operatorname{\mathrm{Re}}G and ReG(0)=∂1w1(0)=0\operatorname{\mathrm{Re}}G(0)=\partial_{1}w_{1}(0)=0 together imply that gg has zero mean. | ReG\operatorname{\mathrm{Re}}G implies that ReG(0)=0\operatorname{\mathrm{Re}}G(0)=0 and hence G(0)=0G(0)=0. It is now | =reiθ+ε(reiθ)m+1+O(ε2),\displaystyle=re^{i\theta}+\varepsilon(re^{i\theta})^{m+1}+O(\varepsilon^{2}), | (Transversality) ∂μL(μ∗)ξ∗∉ranL(μ∗)\partial_{\mu}L(\mu^{*})\xi^{*}\notin\operatorname{\mathrm{ran}}L(\mu^{*}). | B |
In each case, XX will always be clear from context whenever we employ the notation |⋅|∞\left|\cdot\right|_{\infty}. | Regarding the dual space 𝒞(ΩA;𝒜)′\mathcal{C}(\Omega_{A};\mathcal{A})^{\prime}, and the corresponding dual operator | Let XX be a Banach space. We denote its topological dual space by X′X^{\prime}. We also consider X′X^{\prime} | Let XX be a Banach space. We denote its norm by |⋅|X\left|\cdot\right|_{X}. We also consider 𝒞(ΩA;X)\mathcal{C}(\Omega_{A};X) | and denote the set of continuous functions from ΩA\Omega_{A} to a topological space XX by 𝒞(ΩA;X)\mathcal{C}(\Omega_{A};X). | B |
If ℜ\mathfrak{R} is pp-positive, then b1(M)=⋯=bn−p(M)=0b_{1}(M)=\cdots=b_{n-p}(M)=0 and bp(M)=⋯=bn−1(M)=0b_{p}(M)=\cdots=b_{n-1}(M)=0. | Since m≤(n−p+1)(p+q)2(p+1)m\leq\frac{(n-p+1)(p+q)}{2(p+1)}, we have m≤Cp,qkm\leq C_{p,q}^{k} for 0≤k≤t0\leq k\leq t and therefore 𝔹p,q\mathbb{B}^{p,q} is positive. | If q≤p+1q\leq p+1, p≤n2p\leq\frac{n}{2} and m≤n−p+q2m\leq\frac{n-p+q}{2}. By Theorem 1.4, 𝔹p,q\mathbb{B}^{p,q} is semi-positive, and one has H∂¯p,q(M,E)=0H_{\overline{\partial}}^{p,q}(M,E)=0. | If ℜ\mathfrak{R} is pp-positive and 2p≤n2p\leq n, then bk(M)=0b_{k}(M)=0 for all 1≤k≤n−11\leq k\leq n-1. | If ℜ\mathfrak{R} is pp-semipositive, then any harmonic kk-form is parallel for 1≤k≤n−p1\leq k\leq n-p, or p≤k≤n−1p\leq k\leq n-1. | C |
Organization of article: In section 2, we enlist the variational framework and preliminary results. We demonstrate the proof of the concentration compactness principle for the pp-biharmonic operator with critical Choquard-type nonlinearity. In section 3, we discuss the fundamental results for Palais-Smale sequence, for... | Organization of article: In section 2, we enlist the variational framework and preliminary results. We demonstrate the proof of the concentration compactness principle for the pp-biharmonic operator with critical Choquard-type nonlinearity. In section 3, we discuss the fundamental results for Palais-Smale sequence, for... | In this section we recall the main notations and tools that will be needed in the sequel. Define the space | In this section, first we show that the functional is coercive and bounded below and then we prove the Theorems 1.4 and 1.5. | In this paper, we consider the critical Choquard-Kirchhoff equation involving pp-biharmonic operator in the whole space ℝN\mathbb{R}^{N}. Motivated by the results introduced in [25, 18] and a few papers on the pp-biharmonic operator, we established the existence and multiplicity of solutions to the problem (𝒫α,λ)(\mat... | B |
Now combining (32) and (33), we get immediately (30) and the desired result (29) of Theorem 3.3 follows from (30) and (31). This proof completes the proof. | Since the kernel Kℐ(s,x)K_{\mathcal{I}}(s,x) is a finite sum of HH-functions, a direct analysis of its behaviour near zero and infinity would therefore be interesting to expedite. Under the conditions given in (24), we can use Corollary 1.10.1 of [3] to obtain | Thus, the assertions (25) and (27) suggest the imposition of the condition xλ−δφ(x)∈L1(ℝ>0)x^{\lambda-\delta}\varphi(x)\in L^{1}(\mathbb{R}_{>0}) as stated in the hypothesis of Theorem 3.1. | Next, we evaluate K𝒥(s,x)≡ℐxλe−sxK_{\mathcal{J}}(s,x)\equiv\mathcal{I}x^{\lambda}\mathrm{e}^{-sx} involved in (31). Using the integral representation (21), we obtain | As in Remark 3.2, we give a direct analysis of the behaviour of the kernel K𝒥(s,x)K_{\mathcal{J}}(s,x) involved in the integral operator (29). Using the formula [3, p. 11, Eq. (1.5.13)], we have | D |
However, using the negative gradient direction often results in slow convergence, especially for nonconvex functions [9, 19, 31]. Furthermore, in cases where the Hessian matrix exhibits a large condition number, gradient-based methods become inefficient because of excessively slow convergence, as noted in [32, Chapter ... | During the iteration process of PDOME algorithm, constructs and minimizes a majorant function in a hybrid direction based on extrapolation. Theoretical analysis confirms that the algorithm can achieve convergence to a critical point, and its global convergence rate is studied based on the KL property. Numerical experim... | To address the limitations identified in Subsection 2.1, we propose the sPDOME and PDOME algorithm. These approaches integrates a dogleg search strategy, drawing inspiration from trust region methods. At each iteration, the algorithm constructs and minimizes an opportunistically majorized surrogate function along the d... | In this section, we introduce the PDOME algorithm for solving problem (1) which may involve nonconvexity and nonsmoothness, aiming to address the limitations outlined in Subsection 2.1. Building on the existing technical framework, we further propose a novel extrapolation technique to enhance the algorithm’s performanc... | The core idea of PDOME algorithm is based on the majorization-minimization (MM) framework and incorporates extrapolation acceleration techniques. This algorithm constructs a surrogate function along the dogleg path at the extrapolated point, integrating the gradient direction and Newton-type search direction: the gradi... | B |
Eαβγδ=2λμλ+2μaαβaγδ+μ(aαγaβδ+aαδaβγ).E^{\alpha\beta\gamma\delta}=\frac{2\lambda\mu}{\lambda+2\mu}a^{\alpha\beta}a^{\gamma\delta}+\mu(a^{\alpha\gamma}a^{\beta\delta}+a^{\alpha\delta}a^{\beta\gamma}). | Verifying the well-posedness of this reduced variational problem falls outside the scope of this contribution. The interested reader may nevertheless consult [49] for useful hints on the matter. | The variational formulation of shell problems originally stems from simplifications of the variational formulation of three-dimensional elasticity, albeit stated in curvilinear coordinates. Curvilinear coordinates are very natural in this setting and allow separating the in-plane contributions from the out-of-plane (or... | Dirichlet (or type 1) boundary conditions are prescribed along the outer sides of the panel while Neumann (or type 2) boundary conditions are prescribed all along the window’s boundaries. The boundary data (as well as the source term and initial conditions) are computed from the expressions in (5.3). The exact solution... | where λ\lambda and μ\mu are the Lamé constants. In this contribution, we limit ourselves to isotropic materials that cover the majority of industrial applications. For the constitutive relationship of laminates, which are also relevant in the context of shell structures, the interested reader may consult [52, 53]. | A |
Thus each term of the sum on the right-hand side of (2.2) is 0(mod5)0\pmod{5}. This completes the proof | Ramanujan’s own recurrence for τ(n)\tau(n) is (2.2) (with r=−24r=-24); from this he computed 30 values of τ(n)\tau(n) in his seminal paper [11]. His proof uses log derivatives, and is along the lines of our proof of (2.1). Gould [8] credits an equivalent form of (2.1) to Rothe (1793). | Taking log derivatives to obtain recurrences from generating functions is a standard procedure in generatingfunctionology, explained in Chapter 1 of Wilf [12]. It was one of Ramanujan’s favorite tricks. For example, we find the following recurrence in Ramanujan’s work | See also Entries 12 and 13 in Chapter 10 of Ramanujan’s notebooks [3, pages 28–32] for some intricate examples illustrating Ramanujan’s use of log differentiation. | We first prove a recurrence relation satisfied by the coefficients of powers of any generating function. | B |
Let us briefly describe our main idea use in this article. Our approach to this problem is to start with a surface SS fibered over ℙ1{\mathbb{P}}^{1} that admits elliptic fibration over ℚ¯\bar{\mathbb{Q}}. That means that the general fibers of this fibration are elliptic curves. We consider the torison elements of the ... | Let us briefly describe our main idea use in this article. Our approach to this problem is to start with a surface SS fibered over ℙ1{\mathbb{P}}^{1} that admits elliptic fibration over ℚ¯\bar{\mathbb{Q}}. That means that the general fibers of this fibration are elliptic curves. We consider the torison elements of the ... | (I) The notion of Hilbert scheme and the Hom scheme makes sense for an arithmetic variety. This is as explained in [4, Chapter: Hilbert schemes and Quot schemes, §5]. | To accomplish this, we start with the theory of Chow schemes and Hilbert schemes for arithmetic varieties, which parameterize cycles on an arithmetic variety, and then use ’etale monodromy of a smooth fibration to conclude that the torsion element mentioned above vary in a family. In this regard we would like to mentio... | (II) The family of Weil divisors of a smooth fibration over Spec(ℤ){\rm{Spec}}({\mathbb{Z}}) is parameterized by a Chow variety, which is actually given by the Picard scheme parameterizing relative Cartier divisors of the same family [13, Corollary 11.8]. In our case the family is 𝒞U{\mathscr{C}}_{U} which is of fini... | C |
Each example highlights some communication constraints and their influence on the stability of motion and formation of agents. | The velocity information is not transmitted from the virtual leader to the active agents, i.e., h0=h0τ=0h_{0}=h^{\tau}_{0}=0. | Figure 8: Schematic diagram of the coupled system. Active agents R1,…,RNR_{1},\dots,R_{N} exchange information with each other, and receive information from the virtual leader. | We first analyze the case where all active agents receive information only from the virtual leader, which we call the case of uncoupled agents, see Fig. 2. | k0k_{0}, k0τk^{\tau}_{0}, h0h_{0}, and h0τh^{\tau}_{0} determine the coupling between the agents and the virtual leader. V0(t)=R˙0(t)V_{0}(t)=\dot{R}_{0}(t) and U0(t)=R¨0(t)U_{0}(t)=\ddot{R}_{0}(t) are the velocity and acceleration of the “virtual leader”, i.e. a prescribed position in space that should be followed... | A |
For the remainder of this paper, we will refer to Sidon sets in 𝔽2n\mathbb{F}_{2}^{n} as quad caps or just caps. | Note that we are not concerned with odd zero-sum sets, because those do not pertain to the affine geometric structure of 𝔽2n\mathbb{F}_{2}^{n}. | In [2], the authors study caps of dimension 77 in 𝔽2n\mathbb{F}_{2}^{n} by defining the extended cap basis type ([2, Definition 2.8]). They first characterize the equivalence classes of these caps in terms of their possible extended basis types. For each, they construct a template of the cap, consisting of a basis {𝒂... | Although each quad cap has a well-defined cardinality and dimension, these two numbers do not characterize the set, even up to affine equivalence. Quad caps generally do not have any apparent algebraic or geometric structure, and so determining when they are affinely equivalent can be challenging. In this paper, we dev... | For the remainder of this paper, we will refer to Sidon sets in 𝔽2n\mathbb{F}_{2}^{n} as quad caps or just caps. | C |
ω¯V={𝒙|𝒙=𝒙jV,𝒙jV∈Ω¯,j=1,2,…,MV},\overline{\omega}^{V}=\{\bm{x}\ |\ \bm{x}=\bm{x}_{j}^{V},\ \bm{x}_{j}^{V}\in\overline{\Omega},\ j=1,2,\ldots,M_{V}\}, | Conversely, if the solution is defined at the vertices of the Voronoi diagram, then the control volumes coincide with the cells of the Delaunay mesh. | In standard mesh generation, the nodes of the computational mesh are placed at the vertices of the cells. | In the MVD approach, for Delaunay nodes the control volumes are Voronoi cells, while for Voronoi nodes the control volumes are Delaunay cells. | In discretization, mesh cells are defined by their vertices, while in the finite volume method the domain is additionally partitioned into control volumes associated with mesh nodes. | D |
Table 8: The pp-values of paired two-sample tt-tests of the sums of squared errors of ASE with BE and SANVI, respectively, in the example of rank-three latent curve. | We finally apply the proposed algorithm on a real-world network of political blogs (Adamic and Glance,, 2005). The network corresponds to the hyperlinks of blogs regarding U.S. politics after the 2004 presidential election. These blogs are manually classified as either liberal or conservative, which we use as the true ... | The sums of squared errors of the four estimates are summarized in Table 3. We can see that for large samples, the three likelihood-based estimates (OSE, BE, and SANVI) all have smaller sums of squared errors than ASE does, and in the case of n=1000n=1000, the two estimates based on the ESL function still perform well.... | The sums of squared errors of the four estimates are summarized in Table 5. We can see that the errors are obviously larger than those in the previous simulated examples, due to the complex nature of the latent positions obtained from a curve, in comparison to the relatively simple nature of the latent positions in a s... | In this section, we study the finite-sample numerical performance of SANVI in several simulated examples of GRDPG. For comparison, we implement the following competing estimates: ASE, OSE developed by Xie and Xu, (2023), Bayes estimate (BE) as the posterior mean from MCMC with the ESL function, and SANVI. For both BE a... | A |
Wk1,Λ(ℝn)W_{k}^{1,\Lambda}(\mathbb{R}^{n}) denotes Dunkl-Orlicz-Sobolev space (see Definition 3.1). | First, we establish a Caccioppoli-type inequality for the Dunkl-AA-Laplacian, consisting of a generalized differential operator. Specifically, we prove the Dunkl-Caccioppoli-type inequality for functions uu satisfying the DDI. The Dunkl-Caccioppoli-type inequality reads: | The Dunkl-Caccioppoli-type inequality is obtained by first proving the following local estimate (see Proposition 3.3) | Now we are stating the global estimate of Dunkl-Caccioppoli-type inequality, which is the limiting case of the local estimate (3.10). To the best of our knowledge, the Caccioppoli-type inequalities are not derived for the partial differential inequality −ΔA(u)≥bΦ(u)χ{u>0}-\Delta_{A}(u)\geq b\Phi(u)\chi_{\{u>0\}}, w... | From now on, we say that uu satisfies the DDI if u∈Wk,loc1,Λu\in W_{k,\text{loc}}^{1,\Lambda} and the inequality (3.1) holds for all nonnegative compactly supported functions v∈Wk,loc1,Λv\in W_{k,\text{loc}}^{1,\Lambda}. We state our results in the sequel. The proof of the Dunkl-Caccioppoli-type inequality relies on te... | B |
Let (F,∂F)⊂(𝔹4,𝕊3)(F,\partial F)\subset(\mathbb{B}^{4},\mathbb{S}^{3}) be a smooth, oriented, properly embedded symplectic surface; assume that ∂F\partial F is ascending. Then FF is a quasipositive surface. | Suppose that Σn\Sigma_{n}’s, Σ0\Sigma_{0}, MM verify Assumption Main assumption. Assume moreover that the Σn\Sigma_{n}’s are minimal and that kT+kN=0k^{T}+k^{N}=0. | Let MM, (Σn)(\Sigma_{n}) and Σ0\Sigma_{0} be as in Assumption Main assumption and suppose moreover that the Σn\Sigma_{n}’s are minimal. There exists a | where α\alpha is a generic real number which ensures that Σ0\Sigma_{0} is topologically embedded. It follows from [S-V2] that the writhe of ∂Σ0\partial\Sigma_{0} is 2020, up to sign. Thus, if (Σn)(\Sigma_{n}) is a sequence of connected minimal surfaces converging to Σ0\Sigma_{0} as in Assumption 1, we have g(Σn)≥9g(\S... | Let (Σn)(\Sigma_{n}), Σ0\Sigma_{0} be as in Assumption Main assumption and assume that the Σn\Sigma_{n}’s are symplectic for a symplectic structure ω\omega in a neighbourhood of pp. | A |
Mα⊗M~α=0M_{\alpha}\otimes\widetilde{M}_{\alpha}=0 (since the projectors are orthogonal to each other). | All isotropic points 𝔞F{\mathfrak{a}}_{F} of the Balmer spectrum Spc(DMgm(k,𝔽2))\operatorname{Spc}(\operatorname{DM}_{gm}(k,{\mathbb{F}}_{2})) are closed. | We can try to distinguish points of Spc(DMgm(k;𝔽2))\operatorname{Spc}(\operatorname{DM}_{gm}(k;{\mathbb{F}}_{2})) using Rost and reduced Rost motives | to that of DAM(k)\operatorname{DAM}(k). But the varieties of positive dimension can’t be reduced to zero-dimensional ones and the fibers of the projection Spc(DM(k)c)→Spc(DAM(k)c)\operatorname{Spc}(\operatorname{DM}(k)^{c})\rightarrow\operatorname{Spc}(\operatorname{DAM}(k)^{c}) are expected to be “large”. | f∗:DMgm(Y;Λ)→DMgm(Z;Λ).f^{*}:\operatorname{DM}_{gm}(Y;\Lambda)\rightarrow\operatorname{DM}_{gm}(Z;\Lambda). | B |
Suppose that u1,…,un−k∈ℤnu_{1},\ldots,u_{n-k}\in\mathbb{Z}^{n} and v1,…,vm∈ℤnv_{1},\dotsc,v_{m}\in\mathbb{Z}^{n} are vectors satisfying | Then the vector space spanned by v1modp,…,vmmodpv_{1}\bmod{p},\ldots,v_{m}\bmod{p} over 𝔽p\mathbb{F}_{p} has dimension at most k/2k/2. | Suppose that u1,…,un−k∈ℤnu_{1},\ldots,u_{n-k}\in\mathbb{Z}^{n} and v1,…,vm∈ℤnv_{1},\dotsc,v_{m}\in\mathbb{Z}^{n} are vectors satisfying | role is played by the bound |{0,1}n∩V|≤2d\left|\{0,1\}^{n}\cap V\right|\leq 2^{d} that holds for any vector subspace VV of 𝔽pn\mathbb{F}_{p}^{n} of dimension dd. | By removing some of the vectors, we may assume without loss of generality that v1modp,…,vmmodpv_{1}\bmod p,\dotsc,v_{m}\bmod p form a basis for VV, so that m=dimpVm=\dim_{p}V. | A |
The above result shows that these points may be distinguished with the help of pure symbols over the flexible closure of the base field. | Over a field of characteristic zero, the 22-equivalence classes of extensions of the base field parametrize the | describe isotropic points of characteristic 22 of the Balmer spectrum [1] of Voevodsky motives - see [9, Theorem 5.13]. The above permits to show that all these points are | isotropic points of characteristic 22 of the Balmer spectrum Spc(DMgm(k))\operatorname{Spc}(DM_{gm}(k)) [1] of the Voevodsky motivic category - see [9, Theorem 5.13]. | All isotropic points 𝔞2,E{\mathfrak{a}}_{2,E} of characteristic 22 of the Balmer spectrum Spc(DMgm(k))\operatorname{Spc}(DM_{gm}(k)) are | D |
This exceptional property of nonlinear gravitational waves is directly related to the weak form of the | First, her pioneering work [9] on the 3+13+1 decomposition of Einstein’s equations, usually attributed only | crucial time-hyperbolicity condition used by Yvonne and Tommaso Ruggeri, which yields, in 3+1 variables, | the most interesting result of [11] is the realization that the nonlinear structure of Einstein’s equations | Christodoulou-Klainerman “null condition” satisfied by Einstein’s equations (see [14] and references | D |
An unparametrized conformal geodesic is a curve I∋t→γ(t)I\ni t\rightarrow\gamma(t) such that there exists a smooth reparametrization after which the curve becomes a conformal geodesic parametrized by the proper time. | Although the example can be checked directly, it is simpler to use some equivalent formulation of equations of conformal geodesics. | In this paper, we provide an explicit example of a 33-dimensional Riemannian metric with a spiraling conformal geodesic. Although the example is not real-analytic, it can be modified slightly to yield a real-analytic metric with the same property (private communication by Paul Tod). In this way, the original conjecture... | Due to their importance in the study of initial-boundary value problem for general relativity [2, 3] and the character of distinguished curves for conformal geometry [4, 5, 6], this class of curves attracted some attention. It turned out that despite some similarities, many nice properties of standard geodesics do not ... | so c=−|a→|2−u→⋅L^u→c=-|\vec{a}|^{2}-\vec{u}\cdot\hat{L}\vec{u} and the equation is equivalent to equation of conformal geodesics. | A |
When n=4n=4, both E1E_{1} and E2E_{2} have rank equal to 11 (as their determinant is 0, see the matrices H3,…,H10H_{3},\ldots,H_{10}). | The next lemma (see [20, Lemma 8.6]) is frequently used in the theory of group representations. Here it is useful for the proof of part (d) of Theorem 1.2. | In this paper we prove two results. Essentially, the first shows that for circulant Hadamard matrices of order nn, the rank-11 property and properties (a) and (b) in Remark 1.1 hold for n=4n=4 and do not hold when n>4n>4. Our second main result, which follows from a simple characterization of circulant matrices of rank... | Property (b) in Remark 1.1 is significant because of a result of McWilliams [12, Corollary 1.8] (see also Lemma 2.3) which states that for n>2n>2, the orthogonal group 𝕆(n,𝔽2)\mathbb{O}(n,\mathbb{F}_{2}) consists only of the identity matrix. In other words, n=4n=4 is the only case in which (b) holds. | The only circulant, symmetric, and orthogonal matrix over the binary field 𝔽2\mathbb{F}_{2} of order n>2n>2 is the identity matrix InI_{n}. | C |
If x≠zx\neq z, then G−yG-y has a (2,2)(2,2)-AT orientation with respect to (z,x)(z,x), which can be extended to a (2,1)(2,1)-AT orientation of GG with respect to (x,y)(x,y) by orienting the edge incident with yy as an in-arc of yy. | Assume GG is a K4K_{4}-minor-free graph and x,yx,y are distinct vertices of GG. We prove by induction on the number of vertices of GG that GG is (2,2)(2,2)-AT extendable with respect to (x,y)(x,y), and if {x,y}\{x,y\} is not connected by a chain of diamonds, then GG is (2,1)(2,1)-AT extendable with respect to (x,y)(x,y... | We denote by G′G^{\prime} the resulting graph. It follows from the construction that each non-genuine 22-vertex of GG with two 3+3^{+}-neighbors is not a vertex of G′G^{\prime}, and no new 22-vertices are created. In other words, every vertex of G′G^{\prime} has degree at least 33, except genuine 22-vertices of GG or 2... | Thus |B|≥2|B|\geq 2 and n5+=0n_{5}^{+}=0. Then the vertices with odd degree are all 33-vertices, which implies that n3n_{3} is even and so | Thus GG has minimum degree 2, and all vertices other than x,yx,y are 3+3^{+}-vertices. So both x,yx,y are genuine 22-vertices of GG by Lemma 2.7. | D |
⟨χ1,χ2,χ3⟩(δa)={μ2(123)a=3,μ2(231)a=1,0a≠1,3.\langle\chi_{1},\chi_{2},\chi_{3}\rangle(\delta_{a})=\left\{\begin{array}[]{ll}\mu_{2}(123)&a=3,\\ | I would like to express my sincere gratitude to my supervisor, Professor Masanori Morishita, for suggesting the problem studied in this paper and for his guidance and cooperation in carrying out this research. | The purpose of this paper is to introduce the triple quadratic residue symbol [𝔭1,𝔭2,𝔭3][\mathfrak{p}_{1},\mathfrak{p}_{2},\mathfrak{p}_{3}] for certain primes 𝔭i\mathfrak{p}_{i}’s of real quadratic fields and investigate its properties. | In this section, following Morishita’s work ([Mo1], [Mo2], [Mo3]), we introduce mod 2 arithmetic Milnor invariants and triple quadratic residue symbols for certain primes of a real quadratic field, by using the presentation in Theorem 1.3 in Section 1. | For this, we apply theorems due to Höchsmann and Koch ([Ko]) and we assume that the narrow class group of kk is trivial and a certain cohomological obstruction BSB_{S} vanishes, in order to give a precise minimal generators and their relations. | A |
We remark that the variational energy ℰg\mathcal{E}_{g} can also be calculated to third-order accuracy due to its equivalence to μg−β2‖ug‖L4(Ω)4\mu_{g}-\frac{\beta}{2}\|u_{g}\|^{4}_{L^{4}(\Omega)}. | which accounts for the higher order interaction correction to the binary interaction. Here, δ\delta is assumed to be non-negative to guarantee the unique positive ground state [57]. We follow the approach proposed in the work [58] by applying the BEFD with convex-concave splitting of the higher-order interaction term. ... | We solve the cubic-quintic GP equation (21) with β=γ=1\beta=\gamma=1. We choose the quantum pendulum potential V(x,y)=1−cos(2πx2+y2)V(x,y)=1-\cos\left({2\pi}\sqrt{x^{2}+y^{2}}\right) which models an optical lattice trap [55]. Our calculation yields μg=2.432\mu_{g}=2.432, and due to geometric confinement, we note th... | We provide the initial data for the gradient flow by following the approach in [17]. In the case of small β\beta, we use the ground state of the linear Schrodinger problem which can be obtained efficiently. In the case of large β\beta, we consider the Thomas-Fermi approximation where the Laplacian term is ignored to so... | Our proposed BEFD method computes the ground states of the GP equation with the cubic linearity 5, but it can also be readily applied to other variants of the equation that include higher order nonlinear terms. We take as the first example the cubic-quintic GP equation [54] | D |
They also studied equivariant Bar-Natan and Lee homologies for periodic knots. We restate the construction of | Khovanov homology is functorial up to a sign. That is to say, for an ambient isotopy relative to the boundary between cobordism Σ\Sigma and Σ′\Sigma^{\prime} in ℝ3×[0,1]\mathbb{R}^{3}\times[0,1], the maps induced on Khovanov homology are equal up to a sign. | In this section, we prove the theorem 2, which is restated below. The functoriality of Khovanov homology is stated in section 2.3.1. In this section and following sections by ≃\simeq, we mean chain homotopy up to a ±\pm sign. | In section 3, we show that equivariant Khovanov homology is functorial up to a sign. More specifically, | and the results about functoriality of Khovanov homology. Moreover, we restate the construction of equivariant Khovanov homology for periodic links. In section 3, we prove the functoriality of the equivariant Khovanov homology. Lastly, in section 4 we show that the equivariant Khovanov homology obstructs to equivariant... | C |
If 𝒟=M×ℕ\mathcal{D}=M\times\mathbb{N}, we say that (𝒫,𝒢,𝒮)(\mathcal{P},\mathcal{G},\mathcal{S}) is a decomposition for (M,f)(M,f). | The specification play a crucial role to obtain the uniqueness of equilibrium states or measures of maximal entropy. | An invariant measure μ∈ℳf(X)\mu\in\mathcal{M}_{f}(X) is called an equilibrium state of ϕ\phi if μ\mu attain this supremum. When ϕ=0\phi=0, the equilibrium states is called measures of maximal entropy. For a invariant measure μ\mu, we define the pressure of μ\mu as | Following their work, we focus the robustness of the uniqueness of the equilibrium states of partially hyperbolic diffeomorphisms with dominated splittings which its central bundle EcE^{c} can be decomposed into some one-dimensional subbundles. | It has been a long-standing problem to find conditions that guarantee the existence and/or uniqueness of equilibrium states. The theory of uniqueness of equilibrium states of uniformly hyperbolic dynamical systems was developed by Sinai, Ruelle and Bowen in the mid-1970s. Sinai initiated the study [Sin72] in the case o... | A |
Ever since Burnside’s discovery [Bu] of the existence of non-inner class-preserving automorphisms, such automorphisms have attracted attention of finite group theorists (see [Ku] or [Ya] for a recent review). | If L=Lie(G)L=\mathrm{Lie}(G), the Lie algebra of a simply connected Lie group GG, then Der(L)\operatorname{Der}(L) can be naturally identified with the Lie algebra of the Lie group Aut(G)\operatorname{Aut}(G) of automorphisms of GG. Under this identification, iDer(L)\operatorname{iDer}(L) becomes the Lie algebra of... | Almost inner derivations first appeared as a tool for producing non-inner class-preserving automorphisms in the work of Gordon and Wilson [GW] on isospectral deformations of Riemannian manifolds. (Later and independently, they were introduced in [SSB, AS] under the name pointwise inner derivations.) | A systematic algebraic study of almost inner derivations of Lie algebras was initiated in [BDV] (see also [AS]). For finite-dimensional simple Lie algebras, all derivations are inner and therefore they have no nontrivial almost inner derivations. Thus it is not surprising that all known examples of almost inner derivat... | Gordon and Wilson proved that if ϕ:G→G\phi:G\to G is an almost inner automorphism of a solvable Lie group GG with a right-invariant metric gg, then for any cocompact discrete subgroup Γ⊂G\Gamma\subset G, the Riemannian manifolds X=G/ΓX=G/\Gamma and X′=G/ϕ(Γ)X^{\prime}=G/\phi(\Gamma) are isospectral, that is, the Lapla... | B |
HilbG/R:CAlgRcn→𝒮,R′↦Hilb(GR′/R′)\mathrm{Hilb}_{G/R}\!:\mathrm{CAlg}^{\mathrm{cn}}_{R}\to\mathcal{S},\quad R^{\prime}\mapsto\mathrm{Hilb}(G_{R^{\prime}}/R^{\prime}) | To prove relative representability for effective Cartier divisors below, we need the representability of Picard functors. Given a map f:𝖷→Spe´tRf\!:\mathsf{X}\to\mathrm{Sp\acute{e}t}\,R of spectral Deligne–Mumford stacks, we can define a functor | In this subsection, we define relative effective Cartier divisors in the context of spectral algebraic geometry. We then use Lurie’s spectral Artin representability theorem to prove that relative effective Cartier divisors are representable in certain cases. Let us first recall this spectral analogue of Artin’s represe... | The representability of this functor is a special case of [Lur04, Theorem 8.3.3], which we record below. Like that theorem and Theorem 2.17, it can be deduced from the spectral Artin representability theorem 2.10. | We apply Lurie’s spectral Artin representability theorem and verify the 5 criteria from Theorem 2.10 one by one, in the case of n=0n=0, as follows: | C |
To be specific, the path loss w.r.t the BS-TU link can be expressed as ΩSD(ξ)=Ωhd,u,i⋅GS⋅GD\Omega_{\textrm{SD}}(\xi)=\Omega_{h_{d,u,i}}\cdot G_{\text{S}}\cdot G_{\text{D}} with GS(GD)G_{\text{S}}(G_{\text{D}}) being the antenna gain at the BS(TU), and ξ\xi denoting the spatial distance between the source and the dest... | To overcome the destructive effect of multi-path fading, the phase-shifts of the strongest RIS are reconfigured such that the received signals of the dual-hop communications are constructively added to achieve the largest SINR at TU. Mathematically, the optimal phase-shift configuration of the ll-th reflecting element ... | In this section, we first properly select a set of strong RISs based on the physical distances of BS-RIS-nn, and the physical distances of RIS-nn-TU, with n=1,…,Nn=1,\ldots,N. Afterwards, we reconfigure the phase-shifting control of the selected RISs to improve the successful reflective ratio of the dual-hop mmWave com... | Multiple-hop communication with multi-RIS reflections will incur severe multi-path fading, as the cumulative pathloss with respect to the BS-RIS (nn-1)-RIS nn-RIS (nn+1)-UE link will become very big, thus leading to destructive effect of the multi-path fading. Therefore, we only consider dual-hop concept in multi-RIS-a... | reflecting elements, and can not guarantee maximum achievable rate for multiple RIS-aided networks. iii) The proposed optimal RIS phase-shifting control approach is valid for both the multi-RIS-aided systems [6],[20], and multi-hop multi-RIS-aided networks [34]. | C |
ζD:=e2πi/D\zeta_{D}:=e^{2\pi i/D}, (∗∗)(\frac{*}{*}) is the Kronecker symbol, and A(D,d)A(D,d) is the DDth Fourier coefficient of fd=q−d+O(q)f_{d}=q^{-d}+O(q) in the space of weakly holomorphic modular forms of weight 1/2 on Γ0(4)\Gamma_{0}(4) satisfying Kohnen plus condition. | Later in order to describe the generalized Borcherds product explicitly, for instance [4, 6], this alternative infinite product expansion of modular forms was employed. In a similar fashion, we adopt Zagier’s idea of expressing modular forms via this type of infinite product expansion and apply it to all meromorphic mo... | The first purpose of this paper is, as a subsequent work to [16], to provide another description of multiplicative Hecke operators by utilizing an alternative expression for the infinite product expansion of modular forms. Before stating our results, following [23], we first introduce the notion of an alternative expre... | The exponents in the infinite product expansion of a meromorphic modular form are closely related to its divisors. In [5], it was shown that for a given meromorphic modular form ff, the values of the Faber polynomials Jm(τ)J_{m}(\tau) at the zeros or poles of ff in the fundamental domain of SL2(ℤ)\mathrm{SL}_{2}(\mat... | Given another infinite product expansion of a modular form, the action of multiplicative Hecke operators yields not a simple formula for Fourier coefficients of a modular form but rather a natural formula for exponents in its infinite product expansion. It is therefore natural to ask how to compute the exponents in the... | A |
The Fock exchange operator constitutes a cornerstone of HF theory, playing a pivotal role in accurately describing quantum mechanical exchange effects essential for predicting electronic structures in molecules and condensed matter systems. However, its nonlocal nature introduces prohibitive computational costs, partic... | Once the approximate exchange operator is constructed, subsequent computations become computationally trivial. Consequently, efficient realization of these approximate operators becomes paramount for overall performance. To achieve this, we implement a two–level nested SCF iteration strategy: the outer–loop focuses on ... | By isolating the complexity of the Fock exchange operator to the outer loop, this approach provides a robust solution for accelerating HF self–consistency in systems where exchange effects are critical. | This work overcomes these limitations by proposing an adjustable, low–rank approximate exchange operator framework that preserves essential quantum interactions while significantly enhancing computational efficiency as the system size increases. The two–level nested SCF iteration strategy enhances computational efficie... | involves computationally intensive many–electron integrals. By fixing the orbitals that define Fock exchange operator during the inner loop iterations, the outer loop minimizes redundant recalculations of the Fock matrix. This freezing of Fock exchange operator allows the Hamiltonian to depend solely on the electron de... | C |
If η∈𝒫(X,L)𝕋\eta\in\mathcal{P}(X,L)^{\mathbb{T}} realises the equality in (1.8), then η(χ)−1η\eta(\chi)^{-1}\eta is cscS for any minimiser χ∈𝔱+\chi\in\mathfrak{t}_{+} of ℰℋ\mathcal{EH}. The converse holds, when ℰℋmin≤0\mathcal{EH}_{\min}\leq 0. | As a sharp contrast to the aforementioned results on the weighted K-energy, the CR Yamabe energy detects the cscS structures without having to prescribe their Sasaki-Reeb field beforehand. This is a promising advantage to attack open problems regarding the neighborhood of a cscS structure (and eventually the moduli spa... | This last result also implies that the Yamabe energy does not detect the cscS structures that are not minima of the Einstein-Hilbert functional. | This shows that the CR Yamabe energy identifies the cscS structures whose Sasaki-Reeb vector fields minimise ℰℋ\mathcal{EH}. In contrast, we proved in [LahdiliLegendreScarpa] that every cscS structure is a critical point of the Einstein-Hilbert functional, giving a way to study the space of cscS structures on (X,L)(X,... | In this paper, which is a continuation of our previous work [LahdiliLegendreScarpa], we study the CR Yamabe energy of a polarized compact complex manifold (X,L)(X,L). Our motivation is to develop this functional as a tool to study the existence of constant scalar curvature Sasaki (cscS, for brevity) structures on the t... | C |
This paper is organized as follows. In Section 2, we provide some preliminaries that will be used in what follows. | Recall that the dimension of 𝔼1\mathbb{E}_{1} equals the cardinality of S(Λ∗)S(\Lambda^{*}) (see (1.10) in Section 1). The desired result follows from the following two facts: | In Section 3, we give the proof of Theorem 1.6. Section 4 presents an interesting rigidity result, which is independent of the main theorem. | The following energy-enstrophy inequality, which is a direct corollary of the Poincaré inequality, will play an important role in the proof of Proposition 3.1 in Section 3.1. | In this section, we provide an extension of this result, although it is not directly related to the main theorem of this paper. | B |
Let T=e1T1+e2T2∈L1n×eL1n\displaystyle T=e_{1}T_{1}+e_{2}T_{2}\in L_{1}^{n}\times_{e}L_{1}^{n} be a ℂ2\displaystyle\mathbb{C}_{2}-nilpotent operator. Then T1\displaystyle T_{1} and T2\displaystyle T_{2} are singular. | Suppose T\displaystyle T is a ℂ2\displaystyle\mathbb{C}_{2}-nilpotent operator. Then, using ([6], Theorem 3.2.4), ([7], Theorem 1, p.n.590), and Theorem 3.3 we have | Suppose A=e1A−+e2A+\displaystyle A=e_{1}A^{-}+e_{2}A^{+} is a ℂ2\displaystyle\mathbb{C}_{2}-idempotent matrix. Use Definition 4.2 and Theorem 4.4, we have | Suppose A=e1A−+e2A+\displaystyle A=e_{1}A^{-}+e_{2}A^{+} is a ℂ2\displaystyle\mathbb{C}_{2}-idempotent matrix. Then by using Definition 4.2, we have | Suppose T=e1T1+e2T2∈L1n×eL1n\displaystyle T=e_{1}T_{1}+e_{2}T_{2}\in L_{1}^{n}\times_{e}L_{1}^{n} is a ℂ2\displaystyle\mathbb{C}_{2}-nilpotent operator. We use Definition 2.4 and Theorem 3.3 ,we have | A |
Table 2.5 shows that both the cardinality and diameter at least triples between consecutive MSTD sets in this sequence. The exponential growth between the sets generated by this method raises the question of whether a more efficient approach exists for producing the desired sequence. | First, for any set of integers AA, we define the dilation x⋅A≔{xa:a∈A}x\cdot A\coloneqq\{xa:a\in A\}. For all integers x,yx,y such that x≠0x\neq 0, the set | The process for generating such a sequence could be generalized to any MSTD, not just those that start out as subsets of nonnegative integers, due to dilations and translations. | Nathanson later formalized the problem, proved several properties of MSTD sets, provided several methods for constructing MSTD sets, and remarked that MSTD sets should be rare [7]. Surprisingly, Martin and O’Bryant [4] proved that as n→∞n\to\infty, a positive percentage of subsets of {1,2,…,n}\{1,2,\dots,n\} are MSTD, ... | To generate A3A_{3}, we set n=p+2=19n=p+2=19. We choose rr to be as large as possible to minimize the diameter and cardinality of LL and {a∗}−B3\{a^{*}\}-B_{3}. For the same reasons, we choose kk to be as small as possible. We set r=16r=16 and k=2k=2. Then | B |
−(𝐃𝛀x2⊗𝐃xxσt+𝐃𝛀x𝛀y⊗𝐃crossσt+𝐃𝛀y2⊗𝐃yyσt)+𝐈⊗𝚺t-({\bf{D}}_{\boldsymbol{\Omega}_{x}^{2}}\otimes{\bf{D}}^{\sigma_{t}}_{xx}+{\bf{D}}_{\boldsymbol{\Omega}_{x}\boldsymbol{\Omega}_{y}}\otimes{\bf{D}}^{\sigma_{t}}_{\textrm{cross}}+{\bf{D}}_{\boldsymbol{\Omega}_{y}^{2}}\otimes{\bf{D}}^{\sigma_{t}}_{yy})+\mathbf{I}\... | We briefly review SI-DSA for solving RTE. For more details, we refer to review papers [2, 6] and DSA for the second-order formulation of RTE [7]. | Inexact full-rank SI-DSA. To our surprise, we numerically observe that our inexact low-rank SI-DSA algorithm is faster and more robust than its full-rank counterpart. We refer readers to Sec. 4.1 for more details. | We are now ready to formulate the main algorithm—an inexact low-rank variant of SI-DSA—for solving the second-order formulation of the RTE, with the goal of reducing computational cost. | We solve the RTE based on its second-order formulation and refer to the review paper [56] for an overview of numerical methods for second-order RTE. We start by defining the even-parity, ψ+\psi^{+}, and the odd-parity, ψ−\psi^{-}, as | A |
Noticing that the last term in (4.4) coincides with ℱu[u]\mathcal{F}_{u}[u], and that equality holds in the two inequalities of (4.4) at the same time if and only if v=uv=u, the claim was proved. | Now, according to Lemma 2.3, we have that φ^i\hat{\varphi}_{i} converges weakly in D1,p(ℝn)D^{1,p}(\mathbb{R}^{n}) to some function φ^∈D1,p(ℝn)∩L2(ℝn;|y|−1vp1∗−2)\hat{\varphi}\in D^{1,p}(\mathbb{R}^{n})\cap L^{2}(\mathbb{R}^{n};|y|^{-1}v^{p_{1}^{*}-2}), and | Thus, by the assumption that uu and v^\hat{v} are close in D1,p(ℝn)D^{1,p}(\mathbb{R}^{n}), we deduce that | Now, if uu is close to v^=va,1,0\hat{v}=v_{a,1,0} in D1,p(ℝn)D^{1,p}(\mathbb{R}^{n})-norm, it follows from compactness that the minimum of the function | It follows from φ^i⇀φ^\hat{\varphi}_{i}\rightharpoonup\hat{\varphi} in D1,p(ℝn)D^{1,p}(\mathbb{R}^{n}) that, up to a subsequence, | C |
Dynamical localization: There exists a nonempty interval I⊂ΣI\subset\Sigma so that I⊂ΣlocI\subset\Sigma_{loc} and for all ψ∈L2(ℝ3)\psi\in L^{2}(\mathbb{R}^{3}), there is a finite constant C≥0C\geq 0, depending on Pω(I)ψP_{\omega}(I)\psi, so that | see [7] for the dimer model, and [8] for the polymer models. For the two-dimensional random Landau operators, these subintervals are away from the Landau levels near the band edges. To our knowledge, the random δ\delta-interaction model on L2(ℝ3)L^{2}(\mathbb{R}^{3}) is the first example of a three-dimensional, ergodi... | In this paper, we prove that a family of random Schrödinger operators (RSO) on L2(ℝ3)L^{2}(\mathbb{R}^{3}), with random δ\delta-interactions located on ℤ3\mathbb{Z}^{3}, exhibits dynamical delocalization with probability one. This results from the existence of bounded, generalized eigenfunctions at positive energies E... | In our previous work [16], we proved that the RDM has a region of pure point spectrum with probability one and exhibits dynamical localization in the same region under the following assumptions of the single-site probability measure μ\mu. | To our knowledge, this is the first example of a three-dimensional ergodic, random Schrödinger operator that exhibits both dynamical localization in one energy regime, and dynamical delocalization in another regime. Although we proved that the region of dynamical localization is a (possibly a subset of) the spectral lo... | D |
The set of {L2/L1∈ℝ+∖ℚ∣L1,L2∈ℝ}\{L_{2}/L_{1}\in\mathbb{R}_{+}\setminus\mathbb{Q}\mid L_{1},L_{2}\in\mathbb{R}\} such that the flow αL\alpha^{L} on 𝒪2\mathcal{O}_{2} is classified, up to cocycle conjugacy, by the unique real number β\beta given by | It is enough to prove the statement assuming L1L_{1} and L2L_{2} are both positive. The strategy is to show that the dual flow αL^\widehat{\alpha^{L}} falls under the umbrella of [42, Theorem C]. | Since the fixed-point algebra of a flow is invariant under scaling the flow, so are the properties of the flow being equivariantly approximately divisible or equivariantly 𝒵\mathcal{Z}-stable. Replacing the flow αL\alpha^{L} with the flow given by t↦α−tLt\mapsto\alpha_{-t}^{L} if necessary, we can assume that L1L_{1} ... | Recall that the strategy for proving Theorem A is to show that the dual ℝ\mathbb{R}-action has the Rokhlin property generically. We shall first prove that the dual actions of rational quasi-free flows have this property. | Following the notation in Theorem A, the strategy is to show that the ℝ\mathbb{R}-C∗\mathrm{C}^{*}-algebra (𝒪2⋊αLℝ,αL^)(\mathcal{O}_{2}\rtimes_{\alpha^{L}}\mathbb{R},\widehat{\alpha^{L}}) generically falls under the umbrella of [42, Theorem C]. In particular, we will classify the corresponding dual actions. | A |
𝐏Y(X;0)={𝕀,if X=Y,𝟎,otherwise.\mathbf{P}_{Y}(X;0)=\begin{cases}\mathbb{I},&\text{if $X=Y$},\\[4.0pt] | In addition, we derived explicit contour-integral formulas for the transition probabilities, thereby placing this model as a new member of the class of exactly solvable multispecies processes. | Figure 2: Hidden-state representation of the mTASEP dynamics shown in Figure 1. Dashed arrows denote an instantaneous transition. | Then the matrix of transition probabilities 𝐏Y(X;t)\mathbf{P}_{Y}(X;t) admits the contour-integral representation | This standard procedure ensures that the required initial condition is satisfied and yields an explicit integral representation of the transition probabilities. | D |
A primitively polarized K3K3 surface over the field 𝔽\mathbb{F} of genus gg exists, see e.g. [Hu, Ch. 2.4]. A general hyperplane section CC has nonempty, finite Wk+11(C)\mathrm{W}^{1}_{k+1}(C) and Wk+12(C)=∅\mathrm{W}^{2}_{k+1}(C)=\varnothing by [W, Prop. 1, Prop. 4]. By constructing Brill–Noether loci on moving cu... | We recall the following lemma, which is [Ke2, Proposition 2.9]. The result applies directly in our context, as the underlying theory holds for arbitrary algebraically closed fields [Gi]. | Our aim in this paper is to prove the Geometric Syzygy Conjecture for canonical curves of genus g≥4g\geq 4 over algebraically closed fields 𝔽\mathbb{F} of characteristic p≥2g−4p\geq 2g-4. Suppose that ϕ:X⊆ℙ𝔽n\phi\,:\,X\subseteq\mathbb{P}^{n}_{\mathbb{F}}, where 𝔽\mathbb{F} is an algebraically closed field of arbitr... | The Geometric Syzygy Conjecture holds for general curves of genus g≥4g\geq 4 defined over an algebraically closed field 𝔽\mathbb{F} of characteristic p≥2g−4p\geq 2g-4. | Over ℂ\mathbb{C}, this conjecture was proven for g≤8g\leq 8 [vB2] and for general genus [Ke2]. In positive characteristic p>gp>g, the conjecture was proven in the important special case g=2kg=2k and the last syzygy space Kk−1,1(C,ωC)\mathrm{K}_{k-1,1}(C,\omega_{C}) ([W]). The aim of this paper is to prove it in full ... | A |
We consider a prototype linear(ized) system of ODEs describing the dynamics of a population evolving in a periodic environment according to linear birth and transition processes (e.g., death, recovery, change of status). | With reference to this model and following [6, 38, 50, 54] and [23, section 7.9] we recall the definition of the BRN in the time-periodic setting. | We introduce the concept of evolution semigroup associated with an evolutionary system [2, 20, 38, 50, 53]. We first recall basic definitions and results on semigroups of linear operators [25], and then we adapt them to the time-periodic case (2.3). | However, in the time-periodic case the classical definition of the BRN for structured populations must be generalized [23, section 7.9]. | The paper is organized as follows. In section˜2, we introduce a prototype linear population model with time-periodic coefficients. With reference to this model, we illustrate the definition of R0R_{0} in the periodic case, and we illustrate the state-of-the-art on the numerical methods for its computation. In section˜3... | A |
By the connectivity of the 1-skeleton (vertices and edges) of the Farey tessellation, there is a sequence of faces and edges f=f(0),e(1),…,e(k),f(k)=f′f=f^{(0)},e^{(1)},\ldots,e^{(k)},f^{(k)}=f^{\prime} such that e(i)e^{(i)} is incident to f(i−1)f^{(i-1)} and f(i)f^{(i)} for all ii. | For each i∈{1,2}i\in\{1,2\}, by construction, the union of all snake segments 𝒮f\mathcal{S}_{f} such that ω(f)=fi\omega(f)=f_{i} is a single oriented curve 𝐒i\mathbf{S}_{i} that starts at vidv_{\mathrm{id}}. | In the second case, there is some w∈U3w\in U_{3} such that HsnakeH_{\mathrm{snake}} is spanned by the roots α,β\alpha,\beta, where eα,eβe_{\alpha},e_{\beta} are the outgoing edges from vwv_{w}. Let ff be the corresponding face. Then the red edges are those on the path from vidv_{\mathrm{id}} to vwv_{w}, together with a... | By definition, the two edges in f′′f^{\prime\prime} incident to vidv_{\mathrm{id}} have different colors. | By the construction above, there is a sequence of snake segments going backwards from f(j)f^{(j)} to a face f′′f^{\prime\prime} with source vidv_{\mathrm{id}}. | D |
If we want to find the largest 77-distance set by smallest 77 distances in of triangular lattice. We will first take A(0,0),B(0,1)A(0,0),B(0,1), and then locate all points PP with d(P,A)∈{1,3,2,7,3,12,13}d(P,A)\in\{1,\sqrt{3},2,\sqrt{7},3,\sqrt{12},\sqrt{13}\} and d(P,B)∈{1,3,2,7,3,12,13}d(P,B)\in\{1,\sqrt{3},2,\sq... | When m=19m=19, the longest diagonal has length 77, so we would consider the case that the length LL of the diagonal is at least 77. The largest distance in this hexagon is LL, and the second largest is (L−12)2+34=L2−L+1\sqrt{(L-\frac{1}{2})^{2}+\frac{3}{4}}=\sqrt{L^{2}-L+1}. Note that the length (L−12)2+(332)2=L2−L+8\... | If we want to find the largest 77-distance set by smallest 77 distances in of triangular lattice. We will first take A(0,0),B(0,1)A(0,0),B(0,1), and then locate all points PP with d(P,A)∈{1,3,2,7,3,12,13}d(P,A)\in\{1,\sqrt{3},2,\sqrt{7},3,\sqrt{12},\sqrt{13}\} and d(P,B)∈{1,3,2,7,3,12,13}d(P,B)\in\{1,\sqrt{3},2,\sq... | When designing the first method, we assume that the smallest mm distances in triangular lattices will yield the maximum mm-distance set. Therefore, we determine all points that have those distances with both (0,0)(0,0) and (0,1)(0,1). Then we build a graph by using these distances to determine the edges, and find the m... | Finally, we will find the maximum clique in such graph. These points would always have distances one of 1,3,2,7,3,12,131,\sqrt{3},2,\sqrt{7},3,\sqrt{12},\sqrt{13} with each other, with AA, and with BB. Therefore, the maximum clique plus AA and BB would form the largest 77-distance set with the aforementioned distances. | D |
≤exp(10λ2σ4‖A−A∘‖F2).\displaystyle\leq\exp\left(10\lambda^{2}\sigma^{4}\left\|A-\overset{\circ}{A}\right\|_{F}^{2}\right). | To finish the proof, we combine (2.5) and (2.7) with the Cauchy-Schwarz inequality (noting that this is unnecessary if MM is diagonal-free): | where we take c1=2c_{1}=2, c2=1c_{2}=1 if MM is diagonal-free and c1=20c_{1}=20, c2=4c_{2}=4 otherwise. | An approach similar in spirit to ours is due Latała (see the appendix of Barthe & Milman, 2013) in which a decoupling inequality for U-statistics is used. This idea is not dissimilar to decoupling (1.4) using its martingale structure. We proceed to provide details of our direct approach below. | The first term above is easy to analyze, since its just a sum of independent sub-exponential random variables. The second term is a little more tricky, and in the literature a convex decoupling inequality is typically used (Rudelson & Vershynin, 2013). Before we proceed, let us introduce A∘\overset{\circ}{A}, the hollo... | A |
ϵc+(β^ρ¯(x,yx,zx))σ≤ϵc+(β^(ρ¯~(x,yx,zx)+ϵρ))σ≤(βρ¯~(x,yx,zx))σ.\epsilon_{c}+(\hat{\beta}\bar{\rho}(x,y_{x},z_{x}))^{\sigma}\leq\epsilon_{c}+(\hat{\beta}(\tilde{\bar{\rho}}(x,y_{x},z_{x})+\epsilon_{\rho}))^{\sigma}\leq(\beta\tilde{\bar{\rho}}(x,y_{x},z_{x}))^{\sigma}. | Hence, one finds i∈𝒜~LP(yx,zx;x)i\in\widetilde{\cal A}_{{\rm LP}}(y_{x},z_{x};x). The proof is complete. | 𝒜~LP(yx,zx;x):={i∈[q]:c~i(x)≥−(βρ¯~(x,yx,zx))σ}\widetilde{\cal A}_{{\rm LP}}(y_{x},z_{x};x):=\{i\in[q]:\tilde{c}_{i}(x)\geq-(\beta\tilde{\bar{\rho}}(x,y_{x},z_{x}))^{\sigma}\} | 𝒜LP(yx,zx;x):={i∈[q]:ci(x)≥−(βρ¯(x,yx,zx))σ},{\cal A}_{{\rm LP}}(y_{x},z_{x};x):=\{i\in[q]:c_{i}(x)\geq-(\beta\bar{\rho}(x,y_{x},z_{x}))^{\sigma}\}, | Any (yx,zx)(y_{x},z_{x}) solving (10) yields 𝒜LP(yx,zx;x)=𝒜(x∗){\cal A}_{{\rm LP}}(y_{x},z_{x};x)={\cal A}(x_{*}). | A |
The hardware cost of conventional MA systems increases with the number of movable elements, as each requires independently controllable driving components. To reduce hardware cost, more efficient MA array architectures are required. Figure 6(c) shows a movable subarray architecture. Instead of moving each antenna eleme... | Moreover, due to the inherent correlation between the legitimate and eavesdropping channels, transmit beamforming inevitably involves a trade-off between enhancing the desired signal and suppressing information leakage. | It is also feasible to integrate the aforementioned implementation architectures to achieve a more effective hybrid design tailored to specific application scenarios. For instance, in a hybrid MA array configuration, each subarray can adopt a mechanically driven architecture (as illustrated in Figure 6(a)) while incorp... | Nevertheless, the adoption of mechanically driven architectures involves trade-offs related to actuation complexity, energy efficiency, and movement latency, all of which need to be addressed to ensure antennas can quickly and accurately respond to the dynamic requirements for achieving secure communications. | In summary, developing efficient MA architectures for various secure communication systems to achieve a desired trade-off between hardware complexity and security performance is an important problem to solve in practice. | D |
F(1/2+ε;μ)−F(1/2;μ)=|ε|s/s+μ(1/4+ε2)−μ/4=(|ε|p−1/s−μ)ε2⩾0.\displaystyle\begin{aligned} F\left(1/2+\varepsilon;{\mu}\right)-F\left(1/2;{\mu}\right)=\left|\varepsilon\right|^{s}/s+{\mu}\left(1/4+\varepsilon^{2}\right)-{{\mu}}/{4}=\left(|\varepsilon|^{p-1}/s-{\mu}\right)\varepsilon^{2}\geqslant 0.\end{aligned} | cμ¯=max𝐱∈𝔹‖∇f(𝐱)‖∞⩾|∇f(𝐱~)|=μ|νi|⩾cμ>cμ¯.c\overline{{\mu}}=\max_{{\bf x}\in{\mathbb{B}}}\|\nabla f({\bf x})\|_{\infty}\geqslant|\nabla f(\widetilde{{\bf x}})|={\mu}|\nu_{i}|\geqslant c{\mu}>c\overline{{\mu}}. | Problem (3.5) always has a local minimizer 1/21/2 for any given μ>0{\mu}>0. In fact, ∂F(1/2;μ)={0}\partial F({1}/{2};{\mu})=\{0\} for any given μ{\mu}. Moreover, for any sufficiently small |ε||\varepsilon|, we have | This example illustrates that problem (EPM) with a penalty like g(x)=x(1−x)g(x)=x(1-x) may have non-binary local minimizers. In contrast, all local minimizers of the problem with SPFs are binary when μ{\mu} is over a certain threshold. | Problem (3.6) always has binary local minimizers when μ>μ¯=21−s{\mu}>\overline{\mu}=2^{1-s}. In fact, | D |
To extend proposition 3.6 to theorem 1.2, it remains to remove the restriction N(ξ)≥eT−T0N(\xi)\geq\mathrm{e}^{T-T_{0}} and n1(−mu2m2)a1(12logm2)∈ℱ1(Y)n_{1}(-\frac{mu_{2}}{m_{2}})a_{1}(\tfrac{1}{2}\log m_{2})\in\mathcal{F}_{1}(Y) from the left side of (54) and remove the corresponding restrictions on SS on the ... | The number of ξ∈𝒟∩Il−1\xi\in\mathcal{D}\cap I_{l}^{-1} with N(ξ)≤e6(T−T0)N(\xi)\leq\mathrm{e}^{6(T-T_{0})} is O(e6(T−T0))O(\mathrm{e}^{6(T-T_{0})}). | where ν\nu is the restriction of μ\mu to PP and μj,mj,λ\mu_{j},m_{j},\lambda are functions of ξ\xi as in theorem 1.1. | The following lemma is used to remove the condition N(ξ)≥e6(T−T0)N(\xi)\geq\mathrm{e}^{6(T-T_{0})}. | These latter restrictions are easily removed as the measure ν\nu show that the integral increases by Of(e−T0+Y−1)O_{f}(\mathrm{e}^{-T_{0}}+Y^{-1}). | D |
As illustrated in Fig. 1, a three-dimensional (3D) Cartesian coordinate system is established to describe the positions of MAs, where the MA array at the BS resides in the yy-OO-zz plane with its central point placed at the coordinate origin (0,0,0)(0,0,0). Let 𝐭m\mathbf{t}_{m}, 1≤m≤M1\leq m\leq M, denote the position... | Therefore, it is a significant trend to explore the potential advantages of secure communications by using large-aperture antenna arrays. Nevertheless, deploying large-scale antenna arrays in high-frequency bands results in a significantly increase in the Rayleigh distance. As a result, transceivers may fall within the... | Then, by denoting 𝐭=[𝐭1,⋯,𝐭M]\mathbf{t}=[\mathbf{t}_{1},\cdots,\mathbf{t}_{M}] as the position vector of the MA, the channel from the BS to the user is represented by | Since the line-of-sight (LoS) path plays a crucial role compared to non-LoS paths in typical near-field environments over high-frequency bands [10], the LoS channel from the BS to the user is adopted. | We assume that the distance from the BS to both the user and eavesdropper is less than the Rayleigh distance, which is expressed as dR=2D2/λd_{R}=2D^{2}/\lambda, where DD and λ\lambda denote the array aperture and the wavelength, respectively. In other words, both the user and eavesdropper are assumed to be positioned... | C |
As the integral must be finite for finite qc(s)q_{\text{c}}(s), this case is inconsistent as s→0s\to 0. | We lastly mention where the inner halo is in a galactic globular cluster. There is no strict definition to locate the boundary of the inner halo from the core, in the same way as the outer-inner halo boundary. If we use the conventional definition, the inner halo can be limited to qc(s)≲q≲qsq_{\text{c}}(s)\lesssim q\l... | The explicit form of ϵc(s)\epsilon_{\text{c}}(s) and qc(s)q_{\text{c}}(s) are unknown so far. We hence consider four possible cases to determine the time-dependence of qc(s)q_{\text{c}}(s). For small ss or the limit s→0s\to 0, (i) qc(s)q_{\text{c}}(s) is finite, (ii) qc(s)→∞q_{\text{c}}(s)\to\infty, (iii) qc(s)→0... | We simplify the form of the OAFP model to reduce the number of the physical quantities based on two ideas. First, as mentioned in (Henriksen, 2015), the general form of the Buckingham’s Pi theorem may include the translational invariance. The OAFP model is invariant only under the translational transformation of time t... | Case (ii): The relation F∗F^{*} has the following form in the limit qc(s)→∞q_{\text{c}}(s)\to\infty | D |
Our aim is to clarify how these models learn and predict by making explicit the roles played by their internal components. | Broadly speaking, Algorithm 1 de-noises and compresses the training data, into a fixed number of “purified” representative paired samples. It objective is to recover the underlying function by removing the noisy labels and correct the input sample locations for ideal points in the input space. | Our preprocessing step (Algorithm 1) and clustering/attention-training step (Algorithm 2), both subroutines of our training algorithm, can (approximately) recover the correct colour for each superpixel group and efficiently encode it into the keys and values matrices, which then perform the same extraction on out-of-sa... | Our analysis shows that performing this step via Algorithm 1 enables approximate reconstruction of the ground-truth function, even from noisy samples. | Broadly speaking, the trained models decompose into three blocks, each performing a distinct function, following an initial “denoising and summarization data-preprocessing step”. | D |
Applying Theorem 5.3, we deduce that v(X)=α(X0)|X⋅ν|v(X)=\alpha(X_{0})|X\cdot\nu| on B1∖B1/2¯B_{1}\setminus\overline{B_{1/2}}. | A rescaled version of the same argument extends this conclusion to any annulus BR∖BR/2¯B_{R}\setminus\overline{B_{R/2}}, and thus to all of ℝ2\mathbb{R}^{2}. | If the functions ψn\psi_{n} are uniformly bounded in Ww,loc1,2(Br+)W^{1,2}_{{\mathrm{w}},\mathrm{loc}}(B^{+}_{r}), then there exists a subsequence converging weakly in Ww,loc1,2(ℝ+2)W^{1,2}_{{\mathrm{w}},\mathrm{loc}}(\mathbb{R}^{2}_{+}) to a limit function ψ0\psi_{0}, which we refer to as a blow-up limit of ψ\psi at... | ∂Br+(X0):={X=(x,y)∈ℝ2:y≥0 and |X−X0|=r}.\partial B_{r}^{+}(X_{0}):=\{X=(x,y)\in\mathbb{R}^{2}\,:\,y\geq 0\text{ and }|X-X_{0}|=r\}. | When the center is omitted, it is understood to be the origin; thus we write Br:=Br(0)B_{r}:=B_{r}(0) and Br+:=Br+(0)B_{r}^{+}:=B_{r}^{+}(0). | A |
This conjecture is referred to as “Odd Hadwiger’s Conjecture”. It is far stronger than Hadwiger’s conjecture. The case t≤4t\leq 4 was proved by Catlin [4] in 1978. Guenin [13] announced a solution of the case t=5t=5 at a meeting in Oberwolfach in 2005. It remains open for t≥6t\geq 6. For graphs with no odd KtK_{t}-mino... | When a 3K13K_{1}-free graph GG has chromatic number upper-bounded by cω(G)c\omega(G) for some constant cc, we can show that the class of such graphs GG satisfy Odd Hadwiger’s Conjecture. First, we obtain the following result. | In a χ(G)\chi(G)-coloring of GG, each color-class has size at most α(G)\alpha(G). Thus, Hadwiger’s conjecture implies the following weaker version. | In [19], it was shown that Odd Hadwiger’s Conjecture and Conjecture 1.4 are equivalent for graphs with independence number at most two. | One strengthening of Hadwiger’s conjecture is to consider the odd-minor variant. We call that GG has an odd clique minor of size at least tt if there are tt vertex disjoint trees in GG such that every two of them are joined by an edge, in addition, all the vertices of trees can be two-colored in such a way that the edg... | B |
Kurka [8] introduced a classification of one dimensional cellular automata according to sensitivity and equicontinuity of the cellular automaton. | Let FF be a one dimensional cellular automaton with radius r.r. The following properties are equivalent: | its evolution in the central window of radius (2r)d(2r)^{d} under the iteration of the cellular automaton) is equal to one then the cellular automaton cannot have measurable irrational eigenvalues. | Kurka [8] introduced a classification of one dimensional cellular automata according to sensitivity and equicontinuity of the cellular automaton. | Let (Aℤd,F)(A^{\mathbb{Z}^{d}},F) be a cellular automaton with radius rdr^{d}. If FF admit a fully blocking word ww of period p≠1p\neq 1 such that for any integer nn , Fn(w)F^{n}(w) is fully blocking too, then FF has at least a non trivial periodic factor. | A |
Forward update rule via θk+1=θk+hαk\theta^{k+1}=\theta^{k}+h\alpha^{k}. This update rule is effective for nonlinear PDEs when the computed αk\alpha^{k} is small. | Forward update rule via θk+1=θk+hαk\theta^{k+1}=\theta^{k}+h\alpha^{k}. This update rule is effective for nonlinear PDEs when the computed αk\alpha^{k} is small. | at each time step. By Proposition 2.5, this error is small only when the DNN is smooth, the vector αk\alpha^{k} is bounded, and hh is small. This may be difficult to satisfy, especially when FF is nonlinear and the dimension dd is high. | Directly updating the network parameters θ\theta at every time step as in (64) may become impractical in high dimensions or for strongly nonlinear right-hand sides. In these regimes, the DTB coefficients αk\alpha^{k} obtained from the linear system (5) or (11) often exhibit large ℓ2\ell^{2} or ℓ∞\ell^{\infty} norms, wh... | Periodic reset rule, as presented in Proposition 2.4. This approach is recommended when the PDE is nonlinear and the α\alpha dynamics are stiff. | D |
Fix a kk-tuple SS of elements in GG, S=(s1,…,sk)S=(s_{1},\dots,s_{k}) and define the map πS:ℤk→G,a¯=(a1,…,ak)↦πS(a¯)=s1a1…skak\pi_{S}:\mathbb{Z}^{k}\to G,\bar{a}=(a_{1},\dots,a_{k})\mapsto\pi_{S}(\bar{a})=s_{1}^{a_{1}}\dots s_{k}^{a_{k}}. Given a probability measure ψ\psi on ℤk\mathbb{Z}^{k}, we define a symmetric p... | In a later section, we discuss finite convex combinations of such measures when both SS and α\alpha are allowed to vary. This is a significant generalization/variation on results contained in [16, 3]. In [16], the case when SS is a singleton, i.e., S=(s)S=(s), is treated (including convex combinations of such measures)... | The collection of all such measures is denoted by MS,α(G)M_{S,\alpha}(G). The aim of this section is to prove Theorem 1.4. | to those in Theorem 1.4. As discussed in Remark 1.5, since the support of νψ,S\nu_{\psi,S} is no longer restricted to a nilpotent group now, the techniques employed in the proof of Theorem 1.4 are no longer applicable. In fact, in this case, the entire argument relies on the following theorem, the proof of which is pos... | In this section, we will investigate a large family of convex combinations of stable-like probability measures and, more specifically, the properties that enable us to obtain estimates of the convolution powers of such measures. The arguments in this section make heavy use of the results in [16, 4]. The main result pro... | B |
&\leq 10\sqrt{20}\cdot(1\vee C_{1})\cdot\big{(}1\vee\left\lVert g\right\rVert_{L^{\infty}(\mathbf{R}\times\mathbf{R})}^{\frac{1}{2}}\big{)}.\end{aligned} | In particular, we deduce from (83) and Lemma 15, that u(t,⋅)u(t,\cdot) is γn\gamma_{n}-Hölder with constant at most Cγ0CnC_{\gamma_{0}}C_{n}. | Using Lemma 7, for each n∈𝐍n\in\mathbf{N}, we have that u(t,⋅)u(t,\cdot) is γn\gamma_{n}-Hölder with constant at most 10Cn10C_{n}, which is uniformly bounded by the above. | The constant γ¯\overline{\gamma} is given as the fixed point of the map γn↦γn+1\gamma_{n}\mapsto\gamma_{n+1}, that is | Then u(t,⋅)u(t,\cdot) is γ\gamma-Hölder with constant at most 2C(1+22γ−1)2C\big{(}1+\frac{2}{2^{\gamma}-1}\big{)}, uniformly in x∈𝐑dx\in\mathbf{R}^{d}. | B |
In the algebraic approach to quantum field theory, a state is a positive, linear, normalized map from the algebra of observables into the complex numbers. | The class of states that are most straightforward to construct are quasi-free states. A state is said to be quasi-free if it is completely specified by its two-point function. Hence, to construct a quasi-free state, it is sufficient to give a two-point function, often in the form of a bi-distribution, and show that it ... | The condition on the antisymmetric part of the two-point function can be traded for a second positivity condition, which can be proven in the same way as the first positivity condition. As a result, the carefully crafted two-point function pulls back to the two-point function of a well-defined state on the physical spa... | It remains to show the Hadamard property of the Unruh state, i.e. to show that the wavefront set of the two-point function is contained in N+×N−{\pazocal N}^{+}\times{\pazocal N}^{-}, where N±{\pazocal N}^{\pm} is the future/past lightcone. The proof relies on the Propapgation of Singularities theorem [39]. It combines... | The Unruh state is an example of a quasi-free state. It first appeared in the context of the exploration of the evaporation of Schwarzschild black holes [29]. It is constructed such that at past null infinity, it resembles the Minkowski vacuum, while at future null infinity, it contains the thermal radiation expected f... | A |
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