Local Proof Research Articles

Solving two-parameter eigenvalue problems using an alternating method

We present a new approach to compute selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. Our method requires computing generalized eigenvalue problems of the same size as the matrices of the initial two-parameter eigenvalue problem. The method is applicable for right definite problems, possibly after performing an affine transformation. This includes a class of Helmholtz equations when separation of variables is applied. We provide a convergence proof for extremal eigenvalues and empirical evidence along with a local convergence proof for other eigenvalues.

Visualizing Mitochondrial Importability of a Protein Using the Yeast Bi-Genomic Mitochondrial-Split-GFP Strain and an Ordinary Fluorescence Microscope.

Proving with certainty that a GFP-tagged protein is imported inside mitochondria by visualizing its fluorescence emission with an epifluorescence microscope is currently impossible using regular GFP-tagging. This is particularly true for proteins dual localized in the cytosol and mitochondria, which have been estimated to represent up to one third of the established mitoproteomes. These proteins are usually composed of a surpassingly abundant pool of the cytosolic isoform compared to the mitochondrial isoform. As a consequence, when tagged with a regular GFP, the fluorescence emission of the cytosolic isoform will inevitably eclipse that of the mitochondrial one and prevent the detection of the mitochondrial echoform. To overcome this technical limit, we engineered a yeast strain expressing a new type of GFP called Bi-Genomic Mitochondrial-Split-GFP (BiG Mito-Split-GFP). In this strain, one moiety of the GFP is encoded by the mitochondrial DNA while the second moiety of the GFP can be tagged to any nuclear-encoded protein (suspected to be dual localized or bona fide mitochondrial). By doing so, only mitochondrial proteins or echoforms of dual localized proteins, regardless of their organismal origin, trigger GFP reconstitution that can be visualized by regular fluorescence microscopy. The strength of the BiG Mito-Split-GFP system is that proof of the mitochondrial localization of a given protein rests on a simple and effortless microscopy observation.

Designing an indigenous treatment model and evaluating its effectiveness on the symptoms of post-traumatic stress disorder in women and men who survived the Kermanshah earthquake

The current research was conducted with the aim of designing a native treatment model and evaluating its effectiveness on the symptoms of post-traumatic stress disorder. Method: The current research population includes all the women and men who survived the Kermanshah earthquake in the cities of Salas Babajani and Sarpol-Zahab, and after obtaining the codes and compiling and implementing the local treatment model, two experimental and proof groups, 10 people each, a total of 20 people by sampling method were selected and randomly assigned to two experimental and control groups. The post-traumatic stress disorder questionnaire was used for the implementation of the present study. The treatment model consists of 15 sessions that were taught to the clients during fifteen weeks. Each training session of the indigenous treatment model (researcher-made) lasted about 100 minutes. Results: The findings showed that re-experience symptoms (P=0.001, F=12.15), avoidance symptoms (P=0.001, F=17.65), and arousal symptoms (P=0.001, F=11.18) were significant at the level of 0.01. In the experimental group, the mean scores of symptoms of post-traumatic stress disorder decreased in post-test and follow-up stages compared to the control group. The mean difference between post-traumatic stress disorder symptoms in post-test and follow-up stages was 0.05. This finding shows that changes in the mean scores of symptoms of post-traumatic stress disorder remained paparaja after the end of the treatment period. Conclusion: It can be concluded that this method of treatment is effective for people suffering from post-traumatic stress disorder.

A Simple Proof of Locality in Quantum Mechanics

While quantum mechanics allows spooky action at a distance at the level of the wave-function, it also respects locality since there is no instantaneous propagation of real physical effects. We show that this feature can be proved in the standard interpretation of quantum mechanics by a simple general result involving commuting Hermitian operators corresponding to distant (causally disconnected) observables. This is reminiscent of satisfying the locality condition in relativistic quantum field theories.

A multi-scale analysis proof of the power-law localization for random operators on Zd

In this paper we give a multi-scale analysis proof of the power-law localization for random operators on Z d for arbitrary d ≥ 1 .

Collective location statistics release with local differential privacy

Location statistics collective release provides essential information to understand crucial phenomena, including points of interest and movement patterns. Sharing location statistics without compromising users’ privacy is critical. The new standard method for calculating private statistical information is differential privacy. Several research works focus on the centralized environment where individuals share their real data with a trusted curator. To calculate private statistics, the trusted curator then adds carefully measured noise. However, this centralized approach is susceptible to privacy breaches in which, by targeting the trustworthy curator, an adversary accesses the true data. By enabling every individual to perturb their records before reaching the curator, local differential privacy overcomes this form of attack. However, the fundamental challenge is that existing local differential privacy algorithms degrade the privacy guarantee when data collection spans over time. Therefore, a methodology to allocate the privacy budget over multiple timestamps is needed to achieve a collective location statistics release. In this paper, using a sliding window approach, we tackle the issue of releasing location statistics with local differential privacy over multiple timestamps. We develop a privacy budget allocation methodology to release collective location statistics with formal local differential privacy proof. Then, we present an approximation strategy to share the closest private statistics to the current timestamp. This strategy optimizes the released statistics’ utility. We demonstrate our solution enables collective location statistics release with a robust privacy guarantee on two datasets (real-time counts of nearby users and historical counts of bike owners close to each bike station).

An Eigensystem Approach to Anderson Localization for Multi-particle Systems

This paper revisits the proof of Anderson localization for multi-particle systems. We introduce a multi-particle version of the eigensystem multi-scale analysis by Elgart and Klein, which had previously been used for single-particle systems.

Conflict Analysis for MINLP

The generalization of mixed integer program (MIP) techniques to deal with nonlinear, potentially nonconvex, constraints has been a fruitful direction of research for computational mixed integer nonlinear programs (MINLPs) in the last decade. In this paper, we follow that path in order to extend another essential subroutine of modern MIP solvers toward the case of nonlinear optimization: the analysis of infeasible subproblems for learning additional valid constraints. To this end, we derive two different strategies, geared toward two different solution approaches. These are using local dual proofs of infeasibility for LP-based branch-and-bound and the creation of nonlinear dual proofs for NLP-based branch-and-bound, respectively. We discuss implementation details of both approaches and present an extensive computational study, showing that both techniques can significantly enhance performance when solving MINLPs to global optimality. Summary of Contribution: This original article concerns the advancement of exact general-purpose algorithms for solving one of the largest and most prominent problem classes in optimization, mixed integer nonlinear programs (MINLPs). It demonstrates how methods for conflict analysis that learn from infeasible subproblems can be transferred to nonlinear optimization. Further, it develops theory for how nonlinear dual infeasibility proofs can be derived from a nonlinear relaxation. This paper features a thoroughly computational study regarding the impact of conflict analysis techniques on the overall performance of a state-of-the-art MINLP solver when solving MINLPs to global optimality.

Geometric Proof of Riemann Conjecture

This paper proves Riemann conjecture (RH), i.e., that all the zeros in critical region of Riemann ξ -function lie on symmetric line σ =1/2 . Its proof is based on two important properties: the symmetry and alternative oscillation for ξ = u + iv . Denote . Riemann proved that u is real and v ≡ 0 for β =0 (the symmetry). We prove that the zeros of u and v for β > 0 are alternative, so u (t,0) is the single peak. A geometric model was proposed. is called the root-interval of u (t,β) , if |u| > 0 is inside Ij and u = 0 is at its two ends. If |u (t,β)| has only one peak on each Ij, which is called the single peak, else called multiple peaks (it will be proved that the multiple peaks do not exist). The important expressions of u and v for β > 0 were derived. By , the peak u (t,β) will develop toward its convex direction. Besides, ut (t,β) has opposite signs at two ends t = tj , tj+1 of Ij , also does, then there exists some inner point t′ such that v (t′,β) = 0. Therefore {|u|,|v|/β} in Ij form a peak-valley structure such that has positive lower bound independent of t ∈ Ij (i.e. RH holds in Ij ). As u (t,β) does not have the finite condensation point (unless u = const.), any finite t surely falls in some Ij , then holds for any t (RH is proved). Our previous paper “Local geometric proof of Riemann conjecture” (APM, V.10:8, 2020) has two defects, this paper has amended these defects and given a complete proof of RH.

Quantum Hamiltonians with Weak Random Abstract Perturbation. II. Localization in the Expanded Spectrum

We consider multi-dimensional Schr\"odinger operators with a weak random perturbation distributed in the cells of some periodic lattice. In every cell the perturbation is described by the translate of a fixed abstract operator depending on a random variable. The random variables, indexed by the lattice, are assumed to be independent and identically distributed according to an absolutely continuous probability density. A small global coupling constant tunes the strength of the perturbation. We treat analogous random Hamiltonians defined on multi-dimensional layers, as well. For such models we determine the location of the almost sure spectrum and its dependence on the global coupling constant. In this paper we concentrate on the case that the spectrum expands when the perturbation is switched on. Furthermore, we derive a Wegner estimate and an initial length scale estimate, which together with Combes--Thomas estimate allows to invoke the multi-scale analysis proof of localization. We specify an energy region, including the bottom of the almost sure spectrum, which exhibits spectral and dynamical localization. Due to our treatment of general, abstract perturbations our results apply at once to many interesting examples both known and new.

Computer says no: Verdict explainability for runtime monitors using a local proof system

Monitors in Runtime Verification are often constructed as black boxes: they provide verdicts on whether a property is satisfied or violated by the executing system under scrutiny, without much explanation as to why this is the case. In the best of cases, monitors might also return the trace observed, still leaving it up to the user to figure out the logic employed to reach the declared verdict from this trace. In this paper, we propose a local proof system for Linear Temporal Logic—a popular logic used in Runtime Verification—formalising the symbolic deductions within the constraints of Runtime Verification. We prove novel soundness and partial completeness results for this proof system with respect to the original semantics of the logic. Crucially, we show how such a deductive system can be used as a realistic basis for constructing online runtime monitors that provide explanations for their verdicts; we also show the resulting monitor algorithms to satisfy pleasing correctness criteria identified by other works, such as the decidability and incrementality of the analysis and the irrevocability of verdicts. Finally, we relate the expressiveness of the Linear Temporal Logic proof system to existing symbolic analysis techniques used in Runtime Verification.

Parametric Furstenberg Theorem on random products of [formula omitted] matrices

We consider random products of SL(2,R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense Gδ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrödinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.

A quotient of the Lubin\u2013Tate tower II

In this article we construct the quotient mathcal {M}_mathbf {1}/P(K) of the infinite-level Lubin–Tate space mathcal {M}_mathbf {1} by the parabolic subgroup P(K) subset mathrm {GL} _n(K) of block form (n-1,1) as a perfectoid space, generalizing the results of Ludwig (Forum Math Sigma 5:e17, 41, 2017) to arbitrary n and K/{mathbb {Q}} _p finite. For this we prove some perfectoidness results for certain Harris–Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze’s candidate for the mod p Jacquet–Langlands and mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of mathcal {M}_mathbf {1}/P(K) when n=2, and shows that mathcal {M}_mathbf {1}/Q(K) is not perfectoid for maximal parabolics Q not conjugate to P.

Velocity Control of a PMSM Fed by an Inverter-DC/DC Buck Power Electronic Converter

This paper is concerned with velocity control in a permanent magnet synchronous motor (PMSM) when it is fed by an inverter-DC/DC Buck power converter system as power amplifier. We present, for the first time, a formal local asymptotic stability proof to solve this control problem. We stress that this is the first time that this problem is solved for an AC motor. Our control scheme is simple when compared to differential flatness- and backsteping-based proposals in the literature to solve this problem for DC motors. The key for these achievements is the employment of a novel passivity-based approach which takes advantage of the natural energy exchange among the electrical and mechanical subsystems that compose the inverter-DC/DC Buck power converter-PMSM system. The main features of this novel passivity-based approach are summarized in this paper.

Local Geometric Proof of Riemann Conjecture

Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying ζ and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function , , by geometric analysis, which has the symmetry: v=0 if β=0, and basic expression . We show that |u| is single peak in each root-interval of u for fixed β ∈(0,1/2]. Using the slope ut, we prove that v has opposite signs at two end-points of Ij. There surely exists an inner point such that , so {|u|,|v|/β} form a local peak-valley structure, and have positive lower bound in Ij. Because each t must lie in some Ij, then ||ξ|| > 0 is valid for any t (i.e. RH is true). Using the positivity of Lagarias (1999), we show the strict monotone for β > β0 ≥ 0 , and the peak-valley structure is equiva-lent to RH, which may be the geometric model expected by Bombieri (2000). This research follows Liuhui’s methodology: “Computing can detect the un-known and method”.

Canine Cancer: Strategies in Experimental Therapeutics.

Cancer is the most common cause of death in adult dogs. Many features of spontaneously developing tumors in pet dogs contribute to their potential utility as a human disease model. These include similar environmental exposures, similar clonal evolution as it applies to important factors such as immune avoidance, a favorable body size for imaging and serial biopsy, and a relatively contracted time course of disease progression, which makes evaluation of temporal endpoints such as progression free or overall survival feasible in a comparatively short time frame. These criteria have been leveraged to evaluate novel local therapies, demonstrate proof of tumor target inhibition or tumor localization, evaluate potential antimetastatic approaches, and assess the efficacy, safety and immune effects of a variety of immune-based therapeutics. Some of these canine proof of concept studies have been instrumental in informing subsequent human clinical trials. This review will cover key aspects of clinical trials in dogs with spontaneous neoplasia, with examples of how these studies have contributed to human cancer therapeutic development.

Supercuspidal representations of GLn(F) distinguished by a Galois involution

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$, and let $\sigma$ denote its non-trivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $\pi$ of ${\rm GL}_n(F)$, with coefficients in $R$, such that $\pi^{\sigma}\simeq\pi^{\vee}$ (such a representation is said to be $\sigma$-selfdual) we associate a quadratic extension $D/D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times}$. When $\pi$ is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for $\pi$ to be ${\rm GL}_n(F_0)$-distinguished. When the characteristic $\ell$ of $R$ is not $2$, denoting by $\omega$ the non-trivial $R$-character of $F_0^{\times}$ trivial on $F/F_0$-norms, we prove that any $\sigma$-selfdual supercuspidal $R$-representation is either distinguished or $\omega$-distinguished, but not both. In the modular case, that is when $\ell>0$, we give examples of $\sigma$-selfdual cuspidal non-supercuspidal representations which are not distinguished nor $\omega$-distinguished. In the particular case where $R$ is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan-Kable-Tandon.

Well-posedness of semilinear strongly damped wave equations with fractional diffusion operators and $C^0$ potentials on arbitrary bounded domains

We examine the well-posedness of a strongly damped wave equation equipped with fractional diffusion operators. Ranges on the orders of the diffusion operators are determined in connection with global well-posedness of mild solutions or the global existence of weak solutions. Local existence proofs employ either semigroup methods or a Faedo-Galerkin scheme, depending on the type of solution sought. Mild solutions arising from semigroup methods are either analytic or of Gevrey class; the former produce a gradient system. We also determine the critical exponent for the nonlinear term depending on the orders of the fractional diffusion operators. Thanks to the nonlocal presentation of the fractional diffusion operators, we are able to work on arbitrary bounded domains. The nonlinear potential is only assumed to be continuous while satisfying a suitable growth condition.

Large Deviations of the Lyapunov Exponent and Localization for the 1D Anderson Model

The proof of Anderson localization for the 1D Anderson model with arbitrary (e.g. Bernoulli) disorder, originally given by Carmona-Klein-Martinelli in 1987, is based in part on the multi-scale analysis. Later, in the 90s, it was realized that for one-dimensional models with positive Lyapunov exponents some parts of multi-scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle, leading to nonperturbative proofs for 1D quasiperiodic models. In this paper we present a short proof along these lines, for the Anderson model. To prove dynamical localization we also develop a uniform version of Craig-Simon's bound that works in high generality and may be of independent interest.

Variable exponent Calderón's problem in one dimension

We consider one-dimensional Calder\'on's problem for the variable exponent $p(\cdot)$-Laplace equation and find out that more can be seen than in the constant exponent case. The problem is to recover an unknown weight (conductivity) in the weighted $p(\cdot)$-Laplace equation from Dirichlet and Neumann data of solutions. We give a constructive and local uniqueness proof for conductivities in $L^\infty$ restricted to the coarsest sigma-algebra that makes the exponent $p(\cdot)$ measurable.